Problem12_15
2021年全美数学竞赛128Problems
(E) log2 5
9
Problem (2021 AMC 12B Problem 10) Two distinct numbers are selected from the set {1, 2, 3, 4, . . . , 36, 37} so that the sum of the remaining 35 numbers is the product of these two numbers. What is the difference of these two numbers? (A) 5 (B) 7 (C) 8 (D) 9 (E) 10
(A)
1+a+b+c c
(B) 1 + a + b + c
(C)
1+a+b+c c2
(D)
a+b+c c2
(E)
1+a+b+c a+b+c
16
Problem (2021 AMC 12B Problem 17)
Let ABCD be an isosceles trapezoid having parallel bases AB and CD with AB > CD.
7
Problem (2021 AMC 12B Problem 8 / 10B Problem 14)
Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines?
2002 AMC 10B Problems
2002 AMC 10B Problems Problem 1 The ratio 200220032001632• is: （A ）61 （B ）31 （C ）21 （D ）32 （E ）23 Problem 2For the nonzero numbers and define cb a abc c b a ++=),,(Find (2,4,6) . （A ）1 （B ）2 （C ）4 （D ）6 （E ）24Problem 3The arithmetic mean of the nine numbers in the set {9,99,999,9999,…,999999999} is a 9-digit number M, all of whose digits are distinct. The number M does not contain the digit（A ）0 （B ）2 （C ）4 （D ）6 （E ）8Problem 4What is the value of 14)23()14)(23(+--+-x x x x , when x=4 ?（A ）0 （B ）1 （C ）10 （D ）11 （E ）12Problem 5Circles of radius and are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.（A ）π3 （B ）π4 （C ）π6 （D ）π9 （E ）π12Problem 6For how many positive integers is 232+-n n a prime number?（A ）none （B ）one （C ）two （D ）more than two, but finitely many （E ）infinitely manyProblem 7 Let be a positive integer such that n 1713121+++ is an integer. Which of the following statements is not true?（A ）2 divides n （B ）3 divides n （C ）6 divides n （D ）7 divides n （E ）n>84 Problem 8Suppose July of year N has five Mondays. Which of the following must occur five times in the August of year N ? (Note: Both months have 31 days.)（A ）Monday （B ）Tuesday （C ）Wednesday （D ）Thursday （E ）Friday Problem 9Using the letters A,M,O,S and U , we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" U S A M O occupies position（A ）112 （B ）113 （C ）114 （D ）115 （E ）116 Problem 10Supposethat and are nonzero real numbers, and that the equation 02=++b ax x has solutions and . Then the pair (a,b) is（A ）(-2,1) （B ）(-1,2) （C ）(1,-2) （D ）(2,-1) （E ）(4,4) Problem 11The product of three consecutive positive integers is times their sum. What is the sum of the squares?（A ）50 （B ）77 （C ）110 （D ）149 （E ）194 Problem 12For which of the following values of does the equation 621--=--x k x x x have no solution for ?（A ）1 （B ）2 （C ）3 （D ）4 （E ）5 Problem 13Find the value(s) of such that 032128=-+-x y xy is true for all values of .（A ）32 （B ）or 2341- （C ）or 32-41- （D ）23 （E ）or 23-41- Problem 14The number is the square of a positive integer . In decimal representation,the sum of the digits of is （A ）7 （B ）14 （C ）21 （D ）28 （E ）35Problem 15The positive integers A, B, A-B , and A+B are all prime numbers. The sum of these four primes is（A ）even （B ）divisible by 3 （C ）divisible by 5 （D ）divisible by 7 （E ）prime Problem 16For how many integers is nn -20 the square of an integer? （A ）1 （B ）2 （C ）3 （D ）4 （E ）10 Problem 17A regular octagon ABCDEFGH has sides of length two. Find the area of △ADG .（A ）224+ （B ）26+ （C ）234+ （D ）243+ （E ）28+ Problem 18Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?（A ）8 （B ）9 （C ）10 （D ）12 （E ）16Problem 19Suppose that is an arithmetic sequence with 100...10021=+++a a a and 200...200102101=+++a a a . What is the value of（A ）0.0001 （B ）0.001 （C ）0.01 （D ）0.1 （E ）1Problem 20Let a, b, and c be real numbers such that a-7b+8c=4 and 8a+4b-c=7 Then 222c b a +- is（A ）0 （B ）1 （C ）4 （D ）7 （E ）8Problem 21Andy's lawn has twice as much area as Beth's lawn and three times as much as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?（A ）Andy （B ）Beth （C ）Carlos （D ）Andy and Carlos tie for first （E ）All three tieProblem 22m =90º. Let M and N be the midpoints Let △XOY be a right-angled triangle with XOYof the legs OX and OY, respectively.Given XN=19 and YM=22, find XY .（A）24 （B）26 （C）28 （D）30 （E）32Problem 23Let be a sequence of integers such that and for all positive integers and Then is（A）45 （B）56 （C）67 （D）78 （E）89Problem 24Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point vertical feet above the bottom?（A）5 （B）6 （C）7.5 （D）10 （E）15Problem 25When is appended to a list of integers, the mean is increased by . When is appended to the enlarged list, the mean of the enlarged list is decreased by . How many integers were in the original list?（A）4 （B）5 （C）6 （D）7 （E）82002 AMC 10B Solution Problem 1 Problem 2 (C) Problem 3 The following problem is from both the 2002 AMC 12B #1 and 2002 AMC 10B #3, so both problems redirect to this page. We wish to find , or ,or. This does not have the digit 0, so the answer is (A) 0Problem 4 The following problem is from both the 2002 AMC 12B #2 and 2002 AMC 10B #4, so both problems redirect to this page. (D) 11Problem 5A line going through the centers of the two smaller circles also go through the diameter. The length of this line within the circle is 3+3+2+2=10 Because this is the length of the larger circle's diameter, the length of its radius is 5The area of the large circle is 25π, and the area of the two smaller circles is 9π+4π=13π. To find the area of the shaded region, subtract the area of the two smaller circles from the area of the large circle. 25π-13π=12π (E)Problem 6 The following problem is from both the 2002 AMC 12B #3 and 2002 AMC 10B #6, so both problems redirect to this page.Factoring, we get )1)(2(232--=+-n n n n . Either n-1 or n-2 is odd, and the other is even. Their product must yield an even number. The only prime that is even is 2, which is when is 3. The answer is one. (B) . Problem 7 The following problem is from both the 2002 AMC 12B #4 and 2002 AMC 10B #7, so both problems redirect to this page.Since 4241713121=++, 211424114241)14241(lim 0<+<+<+<∞→n n n From which it follows that 114241=+nand n=42. Thus the answer is Problem 8 The following problem is from both the 2002 AMC 12B #8 and 2002 AMC10B #8, so both problems redirect to this page.If there are five Mondays, there are only three possibilities for their dates: (1,8,15,22,29),(2,9,16,23,30) , and (3,10,17,24,31) .In the first case August starts on a Thursday, and there are five Thursdays, Fridays, and Saturdays in August.In the second case August starts on a Wednesday, and there are five Wednesdays, Thursdays, and Fridays in August.In the third case August starts on a Tuesday, and there are five Tuesdays, Wednesdays,and Thursdays in August. The only day of the week that is guaranteed to appear five times is therefore Thursday. D Problem 9 There are "words" beginning with each of the first four letters alphabetically. Fromthere, there are with as the first letter and each of the first three lettersalphabetically. After that, the next "word" is USAMO , hence our answeris . Problem 10 The following problem is from both the 2002 AMC 12B #6 and 2002 AMC 10B #10, so both problems redirect to this page.Solution 1Since 0)())((22=++=++-=--b ax x ab x b a x b x a x , it follows by comparing coefficients that –a-b=a and that ab=bSince b is nonzero, a=1, and -1-b=1, so b=-2 . Thus (a,b)=(1,-2) CSolution 2Another method is to use Vieta's formulas . The sum of the solutions to this polynomial is equal to the opposite of the coefficient, since the leading coefficient is 1; in other words, a+b=-a and the product of the solutions is equal to the constant term (i.e, a*b=b ). Since is nonzero, it follows that and therefore (from the first equation), b=-2a=-2 Problem 11 The following problem is from both the 2002 AMC 12B #7 and 2002 AMC 10B #11, so both problems redirect to this page.Let the three consecutive positive integers be a-1,a,a+1. so , a(a-1)(a+1)=24a , so (a-1)(a+1)=24, 24=4*6 , so a=5. Hence, the sum of the squares is 77654222=++ B. Problem 12The domain over which we solve the equation is .We can now cross-multiply to get rid of the fractions, we get (x-1)(x-6)=(x-k)(x-2)Simplifying that, we get 7x-6=(k+2)x-2k . Clearly for k=5 we get the equation -6=-10 which is never true. The answer is EFor other , one can solve for : k k x 26)5(-=-, hence kk x --=526 . We can easily verify that for none of the other four possible values of is this equal to or , hence there is a solution for in each of the other cases. Problem 13We have 8xy-12y+2x-3=4y(2x-3)+(2x-3)=(4y+1)(2x-3), as (4y+1)(2x-3)=0must be true for all , we must have , hence x=3/2 D .Problem 14Since, 756425364256425)2(585•=•=•=N .Combing the 's and 's gives us,This is 2048 with sixty-four, 's on the end. So, the sum of the digits of is 2+4+8=14 B Problem 15 The following problem is from both the 2002 AMC 12B #11 and 2002 AMC 10B #15, so both problems redirect to this page.Since A-B and A+B must have the same parity , and since there is only one even prime number, it follows that A-B and A+B are both odd. Thus one of A,B is odd and the other even. Since A+B>A>A-B>2, it follows that A (as a prime greater than 2) is odd. Thus B=2, and A-2, A, A+2 are consecutive odd primes. At least one of A-2, A, A+2 is divisible by 3, from which it follows that A-2=3 and A=5. The sum of these numbers is thus 17, a prime, so the answer is prime. E .Problem 16 The following problem is from both the 2002 AMC 12B #12 and 2002 AMC 10B #16, so both problems redirect to this page.Solution 1Let n n x -=202 , with (note that the solutions do not give any additionalsolutions for ). Then rewriting,12022+=x x n . Since , it follows thatdivides . Listing the factors of , we find that x=0,1,2,3 are the only 4 D solutions (respectively yielding n=0,10,16,18 ).Solution 2Forand the fraction is negative, for it is not defined, and forit is between 0 and 1. Thus we only need to examine and . For and we obviously get the squares and respectively.For prime the fraction will not be an integer, as the denominator will not contain the prime in the numerator.This leaves, and a quick substitution shows that out of these onlyn=16 and n=18 yield a square. Problem 17The area of the triangle ADG can be computed as 2AP DG • . We will now find DG and AP . Clearly, PFG is a right isosceles triangle with hypotenuse of length 2, hence PG=2. The same holds for triangle QED and its leg QD . The length of PQ is equal toFE=2 . Hence GD=2+22, and AP=PD=2+2. Then the area of ADG equals 2342)22)(222(2+=++=•AP DG C Problem 18 The following problem is from both the 2002 AMC 12B #14 and 2002 AMC 10B #18, so both problems redirect to this page.For any given pair of circles, they can intersect at most 2 times. Since there are pairs of circles, the maximum number of possible intersections is 6*2=12 . We can construct such a situation as below, so the answer is D .Problem 19Solution 1We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form likeWe can find the value of the common difference every hundred terms. But we forgot that it happens hundred times. So we have a divide the answer by hundred 100/100=1The answer yields us the common difference of every hundred terms. So you has to simply divide the answer by hundred again to find the common difference between one term 1/100=0.01 CSolution 2Adding the two given equations together givesNow, let the common difference be . Notice that, so we merely need to find to get the answer. The formula for an arithmetic sum is ))1(2(21-+n d a n where is the first term, is the number of terms, and is the common difference. Now we use this formula to find a closed form for the first given equation and the sum of the given equations. For the first equation, we have n=100 . Therefore, we have 50(2a 1+99d)=100, or *(1). For the sum of the equations (shown at the beginning of the solution) we have n=200, so 100(2a 1+199d)=300 or *(2) Now we have a system of equations in terms of and . Subtracting (1) from (2) eliminates, yielding 100d=1, and 100112=-=a a d C Solution 3Subtracting the 2 given equations yieldsNow express each a_n in terms of first term a_1 and common difference x between consecutive terms 100))99()199((...))2()102(())()101(())()100((11111111=+-++++-+++-++-+x a x a x a x a x a x a a x a Simplifying and canceling a_1 and x terms gives100x=1Problem 20a+8c=7b+4 and 8a-c=7-4b Squaring both,andare obtained. Adding the two equations and dividing by gives , soProblem 21 The following problem is from both the 2002 AMC 12B #17 and 2002 AMC 10B #21, so both problems redirect to this page.We say Andy's lawn has an area of . Beth's lawn thus has an area of2x , and Carlos's lawn has an area of 3x . We say Andy's lawn mower cuts at a speed of . Carlos's cuts at a speed of 3y , and Beth's cuts at a speed 32y . Each person's lawn is cut at a speed of rate area , so Andy's is cut in yx time, as is Carlos's. Beth's is cut in yx ⨯43, so the first one to finish is Beth. B . Problem 22 The following problem is from both the 2002 AMC 12B #20 and 2002 AMC 10B #22, so both problems redirect to this page.Let OM=x , ON=y . By the Pythagorean Theorem on △XON, MOYrespectively, ,19)2(222=+y x 22222)2(=+y xSumming these gives 8455522=+y x , so 16922=+y x . By the Pythagorean Theorem again, we have222)2()2(XY y x =+ so 26)169(4)(422==+=y x XY BProblem 23Solution 1First of all, write and in terms of a 2.can be represented by in different ways.Since both are equal toyou can set them equal to each other. 15212322+=+a a 32=a Substitute the value of back intoand substitute that intoSolution 2 Substituting into : . Since , . Therefore,and so on until. Adding the Left Hand Sides of all of these equations gives; adding the Right Hand Sides of these equations gives These two expressions must be equal; henceand and Substituting :Thus we have a general formula forand substituting m=12:Problem 24We can let this circle represent the ferris wheel with center and represent the desired point feet above the bottom. Draw a diagram like the one above. We find out is a triangle. That means and the ferris wheel has made of a revolution. Therefore, the time it takes to travel that much of a distance is of a minute, or seconds. The answer is . Alternatively, we could also say that is congruent to by SAS, so is 20, and is equilateral, andProblem 25 Let be the sum of the integers and be the number of elements in the list. Then2002 AMC10Bwe get the equations and . With a little work, the solution is found to be .2002 AMC 10B Answer Key1.E2. C3. A4. D5.E6. B7. E8. D9. D 10.C11.B 12. E 13. D 14. B 15. E 16.D 17.C 18.D 19.C 20.B21.B 22. B 23. D 24. D 25.A。
名词单复数题目60道填空题
名词单复数题目60道填空题一、基础类（20道）1. There is a ____ (book) on the desk.- 答案：book。
解析：“a”是不定冠词，用于修饰单数可数名词，所以这里填book的单数形式。
2. I have two ____ (pen).- 答案：pens。
解析：“two”表示“两个”，修饰可数名词复数，pen的复数形式是pens。
3. There are some ____ (apple) in the basket.- 答案：apples。
解析：“some”既可以修饰可数名词复数，也可以修饰不可数名词，这里修饰可数名词，apple的复数形式是apples。
4. He has a ____ (dog) and two ____ (cat).- 答案：dog, cats。
解析：第一空“a”修饰单数名词dog；第二空“two”修饰可数名词复数，cat的复数形式是cats。
5. My mother bought me a ____ (dress) yesterday.- 答案：dress。
解析：“a”用于单数可数名词前，dress的单数形式就是dress。
6. There are many ____ (student) in the classroom.- 答案：students。
解析：“many”修饰可数名词复数，student的复数形式是students。
7. I can see a ____ (tree) and some ____ (flower) in the garden.- 答案：tree, flowers。
解析：“a”修饰单数tree；“some”修饰可数名词复数，flower的复数形式是flowers。
8. She has a lot of ____ (book).- 答案：books。
解析：“a lot of”既可以修饰可数名词复数也可以修饰不可数名词，这里修饰可数名词，book的复数形式是books。
excelmodule3整数规划01规划的应用
3.相依决策变量--建厂才建仓库， x1>=x3，x2>=x4
4.决策变量 xi = 0,1 (i = 1,2,3,4)
Chap 15-8
例4 连锁店选址
某连锁店计划在城区的东南西北部建店。有10个位置可供参考。每个 位置的预计投资额和利润如表。并有如下条件：
Cost ($/hour) Location 1 Location 2 Location 3 Location 4 Location 5
Machine 1
ቤተ መጻሕፍቲ ባይዱ13
16
12
14
15
Machine 2
15
--
13
20
16
Machine 3
4
7
10
6
7Chap 15-23
指派（分配）问题是0-1规划的特例，也是运 输问题的特例，在指派问题模型中，每一个产地 的提供量和每一个目的地的需求量均为1，即n=m, ai=bi=1 。
Chap 15-18
指派问题一般模型
1.一般模型：
设c 0 : 第i个人完成第j项任务的效率 ij
（时间成本等）
引入x ij
1
0
第i个人完成第j项任务 否则
模型：
min
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s.t
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nn
f
cij
x ij
i 1 j1
n
x ij
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x ij
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每人一项任务
problem的用法
problem的用法problem有问题;难题;习题等意思，那么你知道problem的用法吗?下面跟着店铺一起来学习一下，希望对大家的学习有所帮助!problem的用法大全：problem的用法1：problem作“问题”解,常指客观存在的并有待解决的困难或问题,也可指提出来的疑难问题,还可指数字、事实等方面的问题、习题或思考题。
problem的用法2：problem可用于答语中,与否定词连用,表示“没有问题”。
problem的用法3：problem有时还可以作定语,表示“难对付的,很成问题的”,可修饰物,也可修饰人。
problem的用法例句：1. The plan is good; the problem is it doesn't go far enough.计划不错;问题在于不够深入。
2. I pushed the problem aside; at present it was insoluble.我把问题抛在一边，目前它还无法解决。
3. The letter was short — a simple recitation of their problem.信写得很短——只是简单地说了一下他们的问题。
4. Doctor believed that his low sperm count was the problem.医生认为他的精子数太低是问题所在。
5. You did us a great favour by disposing of that problem.你解决了那个问题，可算是帮了我们一个大忙。
6. Having identified the problem, the question arises of how to overcome it.发现问题后，如何克服它的问题又出现了。
7. The problem is compounded by the medical system here.这儿的医疗体制使问题进一步恶化。
23年12月六级试卷
23年12月六级试卷一、写作（30分钟）题目：The Importance of Lifelong Learning.Directions: For this part, you are allowed 30 minutes to write an essay on the importance of lifelong learning. You should write at least 150 words but no more than 200 words.二、听力理解（30分钟）Section A.Directions: In this section, you will hear three news reports. At the end of each news report, you will hear two or three questions. Both the news report and the questions will be spoken only once. After you hear a question, you must choose the best answer from the four choices marked A), B), C) and D).News Report 1.Questions 1 - 2 are based on the news report you have just heard.1. A) The discovery of a new species in the rainforest.B) The efforts to protect endangered animals in a certain area.C) A new research on the ecological environment of a region.D) The impact of human activities on a local ecosystem.2. A) By setting up more nature reserves.B) By promoting environmental education.C) By restricting industrial development.D) By conducting more scientific research.News Report 2.Questions 3 - 4 are based on the news report you have just heard.3. A) A new technology for energy - saving in buildings.B) The increasing demand for green buildings in urban areas.C) The challenges faced by the construction industry in energy conservation.D) A successful case of sustainable building design.4. A) It reduces the cost of construction materials.B) It improves the indoor air quality.C) It can generate its own electricity.D) It has a unique architectural style.News Report 3.Questions 5 - 7 are based on the news report you have just heard.5. A) The popularity of online shopping during the holiday season.B) The measures taken by e - commerce platforms to ensure delivery.C) The impact of logistics on the development of e - commerce.D) The problems faced by delivery workers during peak times.6. A) Hiring more part - time workers.B) Using advanced sorting technologies.C) Cooperating with local stores for storage.D) Optimizing delivery routes.7. A) It may lead to a decrease in service quality.B) It has increased the competition among delivery companies.C) It has promoted the innovation of logistics models.D) It has caused some disputes between customers and delivery workers.Section B.Directions: In this section, you will hear two long conversations. At the end of each conversation, you will hear four questions. Both the conversation and the questions will be spoken only once. After you hear a question, you must choose the best answer from the four choices marked A), B), C) and D).Conversation 1.Questions 8 - 11 are based on the conversation you have just heard.8. A) Her experience in studying abroad.B) Her plan for future career development.C) Her research project in the university.D) Her participation in an international conference.9. A) It offers more opportunities for international cooperation.B) It has a more flexible curriculum system.C) It can provide better financial support.D) It has a higher reputation in the academic field.10. A) To gain more practical experience.B) To improve her language skills.C) To expand her professional network.D) To learn about different cultures.11. A) Look for internship opportunities in relevant companies.B) Contact professors in her field for advice.C) Prepare relevant materials for application.D) Attend more academic seminars.Conversation 2.Questions 12 - 15 are based on the conversation you have just heard.12. A) The features of a new mobile phone model.B) The development trend of the mobile phone industry.C) The reasons for the popularity of a certain mobile phone brand.D) The impact of mobile phones on people's daily life.13. A) Its high - quality camera.B) Its long - lasting battery.C) Its large - capacity storage.D) Its user - friendly interface.14. A) They are more concerned about the price.B) They prefer mobile phones with simple functions.C) They are easily influenced by advertising.D) They pay more attention to the appearance.15. A) To do market research for a new product.B) To promote a new mobile phone brand.C) To collect feedback from mobile phone users.D) To introduce the latest mobile phone technology.Section C.Directions: In this section, you will hear three passages. At the end of each passage, you will hear three questions. Both the passage and the questions will be spoken only once. After you hear a question, you must choose the best answer from the four choices marked A), B), C) and D).Passage 1.Questions 16 - 18 are based on the passage you have just heard.16. A) The origin and development of traditional Chinese medicine.B) The effectiveness of traditional Chinese medicine in treating diseases.C) The differences between traditional Chinese medicine and Western medicine.D) The modernization process of traditional Chinese medicine.17. A) By using advanced scientific instruments.B) By learning from Western medical theories.C) By relying on a large amount of clinical experience.D) By combining with modern pharmaceutical technology.18. A) It has been widely recognized around the world.B) It still faces some challenges in international promotion.C) It has completely replaced Western medicine in some areas.D) It is mainly used for the treatment of chronic diseases.Passage 2.Questions 19 - 21 are based on the passage you have just heard.19. A) The history of art exhibitions in a certain city.B) The significance of art exhibitions in promoting cultural exchanges.C) The organization and operation of a large - scale art exhibition.D) The influence of modern technology on art exhibitions.20. A) It can attract more visitors from different regions.B) It can display artworks in a more vivid way.C) It can reduce the cost of exhibition organization.D) It can protect artworks from damage.21. A) Through online voting.B) Through on - site questionnaires.C) Through expert evaluation.D) Through social media comments.Passage 3.Questions 22 - 25 are based on the passage you have just heard.22. A) The relationship between diet and health.B) The benefits of a balanced diet.C) The popular diet trends in modern society.D) The impact of junk food on people's health.23. A) It is rich in vitamins and minerals.B) It can help people lose weight quickly.C) It is easy to prepare and convenient to eat.D) It contains a lot of high - quality protein.24. A) Lack of exercise.B) High - stress lifestyle.C) Unhealthy eating habits.D) Genetic factors.25. A) Follow a strict diet plan.B) Increase the intake of fruits and vegetables.C) Do regular exercise.D) Seek professional medical advice.三、阅读理解（40分钟）Section A.Directions: In this section, there is a passage with ten blanks. You are required to select one word for each blank from a list of choices given in a word bank following the passage. Read the passage through carefully before making your choices. Each choice in the bank is identified by a letter. Please mark the corresponding letter for each item on Answer Sheet 2. You may not use any of the words in the bank more than once.The Internet has revolutionized the way we communicate, learn, and do business. It has _(26)_ the world into a global village, where people from different corners of the earth can interact _(27)_ and instantaneously. However, this digital revolution also brings some challenges. One of the major concerns is the issue of online privacy.With the increasing use of the Internet, our personal information is being collected, stored, and sometimes _(28)_ without our knowledge or consent. Companies and organizations often collect data such as our browsing history, shopping preferences, and social media activities to_(29)_ their marketing strategies or improve their services. But this practice may _(30)_ our privacy rights.Another challenge is the spread of false information. The Internet allows anyone to publish and share information, which means that false or misleading news can _(31)_ quickly and reach a large audience. This can have a negative impact on individuals, society, and even international relations.To address these challenges, governments, Internet companies, and users themselves need to take _(32)_ actions. Governments should enact and enforce laws to protect online privacy and regulate the spread of false information. Internet companies should be more _(33)_ about how theycollect and use user data and take measures to prevent the spread of false news. Users should also be more cautious when sharing personal information online and be critical of the information they receive.In conclusion, while the Internet has brought us many benefits, we must also be aware of the _(34)_ it poses and take steps to mitigate them. Only in this way can we fully enjoy the advantages of the digital age while_(35)_ our rights and interests.Word Bank:A) transformed.B) directly.C) violated.D) enhance.E) spread.F) appropriate.G) transparent.H) risks.I) protecting.J) analyzed.Section B.Directions: In this section, you are going to read a passage with ten statements attached to it. Each statement contains information given in one of the paragraphs. Identify the paragraph from which the information is derived. You may choose a paragraph more than once. Each paragraph is marked with a letter. Answer the questions by marking the corresponding letter on Answer Sheet 2.The Future of Higher Education.A) Higher education has always been an important part of society, as it prepares individuals for future careers and contributes to the overall development of a nation. However, in recent years, higher education has faced numerous challenges and is also on the verge of significant changes.B) One of the main challenges is the rising cost of education. Tuition fees have been increasing steadily in many countries, making it difficult for some students, especially those from low - income families, to afford a college education. This has led to a growing student debt problem, which can have a long - term impact on the financial well - being of graduates.C) Another challenge is the changing job market. With the rapid development of technology, the skills required in the workplace are constantly evolving. Higher education institutions need to adapt their curricula to ensure that students are equipped with the relevant skills for the future job market. However, in many cases, there is a gap between what is taught in universities and what is actually needed in the workplace.D) In addition to these challenges, higher education is also facing competition from alternative forms of learning. Online courses and vocational training programs are becoming more popular, as they offer more flexibility and sometimes lower costs compared to traditional collegeeducation. This has forced higher education institutions to re - evaluate their value proposition and find ways to differentiate themselves.E) Despite these challenges, there are also opportunities for higher education in the future. For example, the globalization of education has opened up new possibilities for international cooperation and exchange. Students can now study abroad more easily, and universities can collaborate with institutions from other countries to offer joint programs or conduct research together.F) Technology also presents an opportunity for higher education to improve its teaching and learning methods. Online learning platforms,virtual reality, and artificial intelligence can be used to enhance the educational experience, making it more engaging and effective. For instance, virtual reality can be used to create immersive learning environments for students to study historical events or scientific phenomena.G) To meet the challenges and seize the opportunities, higher education institutions need to make some fundamental changes. They should focus on providing a more personalized learning experience, taking into account the individual needs and interests of students. This can be achieved throughthe use of technology, such as adaptive learning systems that can adjustthe learning content based on the student's progress.H) Another important change is to strengthen the connection between academia and industry. Universities should work more closely with companies to ensure that their research is relevant to the real - world problems and that their students are well - prepared for the job market. This can bedone through internships, industry - sponsored research projects, and the establishment of career centers on campus.I) In conclusion, the future of higher education is full of both challenges and opportunities. Higher education institutions need to be proactive in addressing the challenges and leveraging the opportunities to ensure their long - term survival and success in a rapidly changing world.36. The increasing tuition fees have caused a student debt problem.37. Online courses and vocational training programs are competitors of traditional higher education.38. Technology can be used to create better learning experiences in higher education.39. Higher education institutions should consider students' individual needs when providing learning experiences.40. Universities should cooperate with companies to make their research more practical.41. The globalization of education provides new opportunities for international cooperation in higher education.42. There is a gap between the knowledge taught in universities and the skills required in the workplace.43. Higher education is important for individuals and the development of a nation.44. Higher education institutions need to find ways to distinguish themselves from other learning forms.45. The future of higher education has both challenges and opportunities.Section C.Directions: There are 2 passages in this section. Each passage is followed by some questions or unfinished statements. For each of them there are four choices marked A), B), C) and D). You should decide on the best choice and mark the corresponding letter on Answer Sheet 2.Passage 1.The sharing economy has emerged as a significant economic phenomenon in recent years. It refers to the economic model in which individuals can share their under - utilized assets, such as cars, homes, or skills, with others through digital platforms.One of the most well - known examples of the sharing economy is ride - sharing services like Uber and Lyft. These platforms connect drivers with passengers, allowing people to get around more easily and at a lower cost compared to traditional taxis. The drivers can use their own cars to earn extra income during their spare time, and the passengers can enjoy a more convenient and affordable transportation option.Another example is home - sharing platforms such as Airbnb. Homeowners can list their spare rooms or entire homes on the platform, and travelers can find accommodation that suits their needs and budgets. This has not only provided more choices for travelers but also enabled homeowners to make some extra money.The sharing economy has several benefits. First, it promotes the efficient use of resources. By sharing under - utilized assets, we can reduce waste and make better use of what we already have. Second, it can create new economic opportunities for individuals. People can earn income by sharing their assets or skills, which can be especially helpful for those who are looking for part - time work or extra income. Third, it can also enhance social interaction. When people share their assets, they oftenhave the opportunity to interact with others, which can build stronger communities.However, the sharing economy also faces some challenges. One of the main challenges is regulation. Since the sharing economy operates in a different way from traditional industries, existing regulations may not be suitable for it. For example, ride - sharing services have faced issues such as driver licensing, insurance, and safety regulations. Another challenge is competition. As the sharing economy grows, there is more competition among different platforms and also between the sharing economy and traditional industries. This can lead to price wars and quality issues.46. What is the sharing economy according to the passage?A) An economic model based on the sharing of digital assets.B) An economic model in which people share their unused assets through digital platforms.C) An economic model that focuses on the sharing of skills among individuals.D) An economic model that promotes the sharing of resources within a community.47. What are the benefits of the sharing economy?A) It reduces waste, creates economic opportunities, and enhancessocial interaction.B) It only provides more choices for consumers.C) It helps traditional industries to develop better.D) It mainly focuses on improving transportation and accommodation.48. What are the challenges faced by the sharing economy?A) Lack of users and high operating costs.B) Difficulty in technology innovation and market expansion.C) Problems with regulation and competition.D) Insufficient support from the government and society.49. Which of the following is an example of the sharing economy?A) A traditional taxi company.B) A hotel chain.C) Uber.D) A car rental company that only rents new cars.50. What can be inferred from the passage about the sharing economy?A) It will completely replace traditional industries in the future.B) It needs to find ways to deal with the challenges to develop better.C) It is only suitable for part - time workers.D) It has no impact on the efficient use of resources.Passage 2.Artificial intelligence (AI) has made remarkable progress in recent years and has the potential to revolutionize many industries. One area where AI is having a significant impact is in healthcare.AI can be used to assist doctors in diagnosing diseases. For example, machine - learning algorithms can analyze medical images such as X - rays,CT scans, and MRIs to detect signs of diseases. These algorithms can learn from a large amount of data and can often detect patterns that may be difficult for human doctors to notice. This can lead to more accurate and earlier diagnoses, which can improve the chances of successful treatment.AI can also be used in drug discovery. The process of developing new drugs is long, complex, and expensive. AI can help by predicting the properties of potential drugs and screening out those that are less likely to be effective. This can save time and.。
2024年青海省中考英语真题
青海省2024年初中学业水平考试英 语Be an all-around developed person.注意事项：1. 本试卷满分120分，考试时间120分钟。
2. 本试卷为试题卷，请将答案写在答题卡上，否则无效。
3. 答卷前请将密封线内的项目填写清楚。
第I 卷（听力及选择题）I. 听力测试（共四部分，共20小题，每小题1分，满分20分）第一部分：听句子，选择相对应的图片。
每个句子仅读一遍。
1. A. B. C.2. A. B. C.3. A. B. C.4. A. B. C.5. A. B. C.第二部分：听句子，选择最佳答语。
每个句子读两遍。
6. A. She is tall. B. She is watching TV. C. She is my sister.7. A. Yes, I can. B. I like music. C. I don't join the music club.8. A. It's black.B. It's 7 dollars.C. It's too big. 9 A. It doesn't matter. B. Not bad.C. I'm really sorry.10. A. Thank you. B. Take it easy.C. No, I'm not beautiful.第三部分：听对话，选择最佳答案。
每段对话读两遍。
11. How did Tom go to school today? A. By bus.B. By bike.C. On foot..12. What does the girl think of being a teacher?A. Relaxing.B. Interesting and meaningful.C. Boring.13. Why do Jack and David look the same?A. Because they are friends.B. Because they are pen pals.C. Because they are twins.14. What will they make?A. Banana milk shake.B. Banana pie.C. Fruit salad.15. Who are the two speakers?A. Teacher and student.B. Classmates.C. Mother and son.第四部分：听短文，根据短文内容填空。
期权期货与其他衍生产品第九版课后习题与答案Chapter (15)
CHAPTER 15The Black-Scholes-Merton ModelPractice QuestionsProblem 15.1.What does the Black –Scholes –Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about theprobability distribution of the continuously compounded rate of return on the stock during the year?The Black –Scholes –Merton option pricing model assumes that the probability distribution of the stock price in 1 year (or at any other future time) is lognormal. It assumes that thecontinuously compounded rate of return on the stock during the year is normally distributed.Problem 15.2.The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?The standard deviation of the percentage price change in time t ∆is where σ is the volatility. In this problem 03=.and, assuming 252 trading days in one year, 12520004t ∆=/=. so that 00019=.=. or 1.9%.Problem 15.3.Explain the principle of risk-neutral valuation.The price of an option or other derivative when expressed in terms of the price of theunderlying stock is independent of risk preferences. Options therefore have the same value in a risk-neutral world as they do in the real world. We may therefore assume that the world is risk neutral for the purposes of valuing options. This simplifies the analysis. In a risk-neutral world all securities have an expected return equal to risk-free interest rate. Also, in arisk-neutral world, the appropriate discount rate to use for expected future cash flows is the risk-free interest rate.Problem 15.4.Calculate the price of a three-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum.In this case 050S =, 50K =, 01r=., 03σ=., 025T =., and12102417000917d d d ==.=-.=.The European put price is 0102550(00917)50(02417)N e N -.⨯.-.--.0102550046345004045237e -.⨯.=⨯.-⨯.=.or $2.37.Problem 15.5.What difference does it make to your calculations in Problem 15.4 if a dividend of $1.50 is expected in two months?In this case we must subtract the present value of the dividend from the stock price before using Black –Scholes-Merton. Hence the appropriate value of 0S is01667010501504852S e -.⨯.=-.=.As before 50K =, 01r =., 03σ=., and 025T =.. In this case12100414001086d d d ==.=-.=-.The European put price is 0102550(01086)4852(00414)N e N -.⨯..-.-. 010255005432485204835303e -.⨯.=⨯.-.⨯.=. or $3.03.Problem 15.6.What is implied volatility? How can it be calculated?The implied volatility is the volatility that makes the Black –Scholes-Merton price of an option equal to its market price. The implied volatility is calculated using an iterativeprocedure. A simple approach is the following. Suppose we have two volatilities one too high (i.e., giving an option price greater than the market price) and the other too low (i.e., giving an option price lower than the market price). By testing the volatility that is half way between the two, we get a new too-high volatility or a new too-low volatility. If we search initially for two volatilities, one too high and the other too low we can use this procedure repeatedly to bisect the range and converge on the correct implied volatility. Other more sophisticated approaches (e.g., involving the Newton-Raphson procedure) are used in practice.Problem 15.7.A stock price is currently $40. Assume that the expected return from the stock is 15% and its volatility is 25%. What is the probability distribution for the rate of return (with continuous compounding) earned over a two-year period?In this case 015=.μ and 025=.σ. From equation (15.7) the probability distribution for the rate of return over a two-year period with continuous compounding is:⎪⎪⎭⎫⎝⎛-ϕ225.0,225.015.022i.e.,)03125.0,11875.0(ϕThe expected value of the return is 11.875% per annum and the standard deviation is 17.7%per annum.Problem 15.8.A stock price follows geometric Brownian motion with an expected return of 16% and a volatility of 35%. The current price is $38.a) What is the probability that a European call option on the stock with an exercise price of $40 and a maturity date in six months will be exercised?b) What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised?a) The required probability is the probability of the stock price being above $40 in six months time. Suppose that the stock price in six months is T S⎥⎦⎤⎢⎣⎡⨯⎪⎪⎭⎫ ⎝⎛-+5.035.0,5.0235.016.038ln ~ln 22ϕT Si.e.,()2247.0,687.3~ln ϕT SSince ln 403689=., we require the probability of ln(S T )>3.689. This is3689368711(0008)0247N N .-.⎛⎫-=-. ⎪.⎝⎭Since N(0.008) = 0.5032, the required probability is 0.4968.b) In this case the required probability is the probability of the stock price being less than $40 in six months time. It is10496805032-.=.Problem 15.9.Using the notation in the chapter, prove that a 95% confidence interval for T S is between22(2)196(2)19600andT T S e S e -/-.-/+.μσμσFrom equation (15.3):⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-+T T S S T 220,2ln ~ln σσμϕ95% confidence intervals for ln T S are therefore20ln ()1962S T +--.σμand20ln ()1962S T +-+.σμσ95% confidence intervals for T S are therefore2200ln (2)196ln (2)196and S T S T e e +-/-.+-/+.μσμσi.e.22(2)196(2)19600andT T S e S e -/-.-/+.μσμσProblem 15.10.A portfolio manager announces that the average of the returns realized in each of the last 10 years is 20% per annum. In what respect is this statement misleading?This problem relates to the material in Section 15.3. The statement is misleading in that a certain sum of money, say $1000, when invested for 10 years in the fund would have realized a return (with annual compounding) of less than 20% per annum.The average of the returns realized in each year is always greater than the return per annum (with annual compounding) realized over 10 years. The first is an arithmetic average of the returns in each year; the second is a geometric average of these returns.Problem 15.11.Assume that a non-dividend-paying stock has an expected return of μ and a volatility of σ. An innovative financial institution has just announced that it will trade a derivative that pays off a dollar amount equal to ln S T at time T where T S denotes the values of the stock price at time T .a) Use risk-neutral valuation to calculate the price of the derivative at time t in term of the stock price, S, at time tb) Confirm that your price satisfies the differential equation (15.16)a) At time t , the expected value of ln S T is from equation (15.3) 2ln (/2)()S T t μσ+--In a risk-neutral world the expected value of ln S T is therefore2ln (/2)()S r T t σ+--Using risk-neutral valuation the value of the derivative at time t is()2[ln (/2)()]r T t e S r T t σ--+--b) If()2[ln (/2)()]r T t f e S r T t σ--=+--then()()2()2()2()22[ln (/2)()]/2r T t r T t r T t r T t fre S r T t e r tf e S S f e S S σσ--------∂=+----∂∂=∂∂=-∂The left-hand side of the Black-Scholes-Merton differential equation is()222()2ln (/2)()(/2)/2ln (/2)()r T t r T t e r S r r T t r r e r S r r T t rfσσσσ----⎡⎤+----+-⎣⎦⎡⎤=+--⎣⎦=Hence the differential equation is satisfied.Problem 15.12.Consider a derivative that pays off n T S at time T where T S is the stock price at that time. When the stock pays no dividends and its price follows geometric Brownian motion, it can be shown that its price at time t ()t T ≤ has the form()n h t T S ,where S is the stock price at time t and h is a function only of t and T .(a) By substituting into the Black –Scholes –Merton partial differential equation derive an ordinary differential equation satisfied by ()h t T ,.(b) What is the boundary condition for the differential equation for ()h t T ,? (c) Show that2[05(1)(1)]()()n n r n T t h t T eσ.-+--,= where r is the risk-free interest rate and σ is the stock price volatility.If ()()n G S t h t T S ,=, then n t G t h S ∂/∂=, 1n G S hnS -∂/∂=, and 222(1)n G S hn n S -∂/∂=- where t h h t =∂/∂. Substituting into the Black –Scholes –Merton differential equation we obtain21(1)2t h rhn hn n rh σ++-=The derivative is worth n S when t T =. The boundary condition for this differential equation is therefore ()1h T T ,= The equation2[05(1)(1)]()()n n r n T t h t T e σ.-+--,=satisfies the boundary condition since it collapses to 1h = when t T =. It can also be shown that it satisfies the differential equation in (a). Alternatively we can solve the differential equation in (a) directly. The differential equation can be written21(1)(1)2t h r n n n h σ=----The solution to this is21ln [(1)(1)]()2h r n n n T t σ=-+--or2[05(1)(1)]()()n n r n T t h t T e σ.-+--,=Problem 15.13.What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is three months?In this case 052S =, 50K =, 012r =., 030=.σ and 025T =..212105365003865d d d ==.=-.=. The price of the European call is 01202552(05365)50(03865)N e N -.⨯..-. 00352070425006504e -.=⨯.-⨯. 506=. or $5.06.Problem 15.14.What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is six months?In this case 069S =, 70K =, 005r =., 035=.σ and 05T =..212101666000809d d d ==.=-.=-. The price of the European put is 0050570(00809)69(01666)e N N -.⨯..--. 002570053236904338e -.=⨯.-⨯.640=. or $6.40.Problem 15.15.Consider an American call option on a stock. The stock price is $70, the time to maturity is eight months, the risk-free rate of interest is 10% per annum, the exercise price is $65, and the volatility is 32%. A dividend of $1 is expected after three months and again after sixmonths. Show that it can never be optimal to exercise the option on either of the two dividend dates. Use DerivaGem to calculate the price of the option.Using the notation of Section 15.12, 121D D ==, 2()0101667(1)65(1)107r T t K e e ---.⨯.-=-=., and 21()01025(1)65(1)160r t t K e e ---.⨯.-=-=.. Since 2()1(1)r T t D K e --<- and 21()2(1)r t t D K e --<-It is never optimal to exercise the call option early. DerivaGem shows that the value of the option is 1094..Problem 15.16.A call option on a non-dividend-paying stock has a market price of $2.50. The stock price is $15, the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is 5% per annum. What is the implied volatility?In the case 25c =., 015S =, 13K =, 025T =., 005r =.. The implied volatility must becalculated using an iterative procedure.A volatility of 0.2 (or 20% per annum) gives 220c =.. A volatility of 0.3 gives 232c =.. A volatility of 0.4 gives 2507c =.. A volatility of 0.39 gives 2487c =.. By interpolation the implied volatility is about 0.396 or 39.6% per annum.The implied volatility can also be calculated using DerivaGem. Select equity as theUnderlying Type in the first worksheet. Select Black-Scholes European as the Option Type. Input stock price as 15, the risk-free rate as 5%, time to exercise as 0.25, and exercise price as 13. Leave the dividend table blank because we are assuming no dividends. Select the button corresponding to call. Select the implied volatility button. Input the Price as 2.5 in the second half of the option data table. Hit the Enter key and click on calculate. DerivaGem will show the volatility of the option as 39.64%.Problem 15.17.With the notation used in this chapter (a) What is ()N x '?(b) Show that ()12()()r T t SN d Ke N d --''=, where S is the stock price at time t21d =22d =(c) Calculate 1d S ∂/∂ and 2d S ∂/∂. (d) Show that when()12()()r T t c SN d Ke N d --=-()21()(r T t c rKe N d SN d t --∂'=--∂ where c is the price of a call option on a non-dividend-paying stock. (e) Show that 1()c S N d ∂/∂=.(f) Show that the c satisfies the Black –Scholes –Merton differential equation.(g) Show that c satisfies the boundary condition for a European call option, i.e., thatmax(0)c S K =-, as t tends to T.(a) Since ()N x is the cumulative probability that a variable with a standardized normal distribution will be less than x , ()N x ' is the probability density function for a standardized normal distribution, that is,22()x N x -'=(b)12()(N d N d ''=+2221()22d d T t σσ⎡⎤=---⎢⎥⎣⎦221()exp ()2N d d T t σσ⎡⎤'=--⎢⎥⎣⎦Because22d =it follows that()21exp ()2r T t Ked T t S σσ--⎡⎤--=⎢⎥⎣⎦As a result()12()()r T t SN d Ke N d --''=which is the required result.(c)22212S K d σσ==Hence1d S ∂=∂ Similarly222d σ=and2d S ∂=∂ Therefore:12d d S S∂∂=∂∂(d)()12()()12122()()()()()r T t r T t r T t c SN d Ke N d d d c SN d rKe N d Ke N d t t t------=-∂∂∂''=--∂∂∂ From (b):()12()()r T t SN d Ke N d --''=Hence()1221()()r T t d d c rKe N d SN d t tt --∂∂∂⎛⎫'=-+- ⎪∂∂∂⎝⎭ Since12d d -=12(d d t t t∂∂∂-=∂∂∂=Hence()21()(r T t c rKe N d SN d t --∂'=--∂(e) From differentiating the Black –Scholes –Merton formula for a call price weobtain()12112()()()r T t d d cN d SN d Ke N d S S dS--∂∂∂''=+-∂∂ From the results in (b) and (c) it follows that1()cN d S ∂=∂(f) Differentiating the result in (e) and using the result in (c), we obtain21121()(d cN d S SN d ∂∂'=∂∂'= From the results in d) and e)222()2122211()121()(21()(2[()()]r T t r T t c c c rS S rKe N d SN d t S S rSN d S N d r SN d Ke N d rcσσ----∂∂∂'++=--∂∂∂'++=-=This shows that the Black –Scholes –Merton formula for a call option does indeed satisfy the Black –Scholes –Merton differential equation(g) Consider what happens in the formula for c in part (d) as t approaches T . IfS K >, 1d and 2d tend to infinity and 1()N d and 2()N d tend to 1. If S K <, 1d and 2d tend to zero. It follows that the formula for c tends to max(0)S K -,.Problem 15.18.Show that the Black –Scholes –Merton formulas for call and put options satisfy put –call parity.The Black –Scholes –Merton formula for a European call option is 012()()rT c S N d Ke N d -=- so that 012()()rT rT rT c Ke S N d Ke N d Ke ---+=-+ or 012()[1()]rT rT c Ke S N d Ke N d --+=+- or 012()()rT rT c Ke S N d Ke N d --+=+-The Black –Scholes –Merton formula for a European put option is 201()()rT p Ke N d S N d -=--- so that 02010()()rT p S Ke N d S N d S -+=---+ or 0201()[1()]rT p S Ke N d S N d -+=-+-- or 0201()()rT p S Ke N d S N d -+=-+ This shows that the put –call parity result 0rT c Ke p S -+=+ holds.Problem 15.19.A stock price is currently $50 and the risk-free interest rate is 5%. Use the DerivaGemsoftware to translate the following table of European call options on the stock into a table of implied volatilities, assuming no dividends. Are the option prices consistent with the assumptions underlying Black –Scholes –Merton?Using DerivaGem we obtain the following table of implied volatilitiesBlack-Scholes European as the Option Type. Input stock price as 50, the risk-free rate as 5%, time to exercise as 0.25, and exercise price as 45. Leave the dividend table blank because we are assuming no dividends. Select the button corresponding to call. Select the implied volatility button. Input the Price as 7.0 in the second half of the option data table. Hit theEnter key and click on calculate. DerivaGem will show the volatility of the option as 37.78%. Change the strike price and time to exercise and recompute to calculate the rest of the numbers in the table.The option prices are not exactly consistent with Black –Scholes –Merton. If they were, the implied volatilities would be all the same. We usually find in practice that low strike price options on a stock have significantly higher implied volatilities than high strike price options on the same stock. This phenomenon is discussed in Chapter 20.Problem 15.20.Explain carefully why Black’s approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Does the answer given by Black’s approach understate or overstate the true option value? Explain your answer.Black’s approach in effect assumes that the holder of option must decide at time zero whether it is a European option maturing at time n t (the final ex-dividend date) or a European optionmaturing at time T . In fact the holder of the option has more flexibility than this. The holder can choose to exercise at time n t if the stock price at that time is above some level but nototherwise. Furthermore, if the option is not exercised at time n t , it can still be exercised attime T .It appears that Black ’s approach should understate the true option value. This is because the holder of the option has more alternative strategies for deciding when to exercise the option than the two strategies implicitly assumed by the approach. These alternative strategies add value to the option.However, this is not the whole story! The standard approach to valuing either an American or a European option on a stock paying a single dividend applies the volatility to the stock price less the present value of the dividend. (The procedure for valuing an American option is explained in Chapter 21.) Black’s approach when considering exercise just prior to the dividend date applies the volatility to the stock price itself. Black’s approach thereforeassumes more stock price variability than the standard approach in some of its calculations. In some circumstances it can give a higher price than the standard approach.Problem 15.21.Consider an American call option on a stock. The stock price is $50, the time to maturity is 15 months, the risk-free rate of interest is 8% per annum, the exercise price is $55, and the volatility is 25%. Dividends of $1.50 are expected in 4 months and 10 months. Show that it can never be optimal to exercise the option on either of the two dividend dates. Calculate the price of the option.With the notation in the text12121500333308333125008and 55D D t t T r K ==.,=.,=.,=.,=.=2()00804167155(1)180r T t K e e ---.⨯.⎡⎤⎢⎥⎣⎦-=-=. Hence2()21r T t D K e --⎡⎤⎢⎥⎣⎦<- Also:21()00805155(1)216r t t K e e ---.⨯.⎡⎤⎢⎥⎣⎦-=-=. Hence:21()11r t t D K e --⎡⎤⎢⎥⎣⎦<- It follows from the conditions established in Section 15.12 that the option should never be exercised early.The present value of the dividends is033330080833300815152864e e -.⨯.-.⨯..+.=.The option can be valued using the European pricing formula with:05028644713655025008125S K r T σ=-.=.,=,=.,=.,=.212100545003340d d d ==-.=-.=-.12()04783()03692N d N d =.,=.and the call price is00812547136047835503692417e -.⨯..⨯.-⨯.=. or $4.17.Problem 15.22.Show that the probability that a European call option will be exercised in a risk-neutral world is, with the notation introduced in this chapter, 2()N d . What is an expression for the value of a derivative that pays off $100 if the price of a stock at time T is greater than K ?The probability that the call option will be exercised is the probability that T S K > where T S is the stock price at time T . In a risk neutral world[]T T r S S T 220,)2/(ln ~ln σσϕ-+The probability that T S K > is the same as the probability that ln ln T S K >. This is21N ⎡⎤-2N =2()N d =The expected value at time T in a risk neutral world of a derivative security which pays off $100 when T S K > is therefore2100()N dFrom risk neutral valuation the value of the security at time zero is2100()rT e N d -Problem 15.23.Use the result in equation (15.17) to determine the value of a perpetual American put option on a non-dividend-paying stock with strike price K if it is exercised when the stock price equals H where H<K. Assume that the current stock price S is greater than H. What is the value of H that maximizes the option value? Deduce the value of a perpetual American put option with strike price K.If the perpetual American put is exercised when S=H , it provides a payoff of (K −H ). We obtain its value, by setting Q=K−H in equation (15.17), as 22/2/2)()(σσ-⎪⎭⎫ ⎝⎛-=⎪⎭⎫ ⎝⎛-=r r S H H K H S H K VNow1/222/222222/21/22/2222222)(212)(212-σσσ-σσ⎪⎭⎫ ⎝⎛σ⎪⎭⎫ ⎝⎛σ-+-+⎪⎭⎫ ⎝⎛σ-=⎪⎭⎫ ⎝⎛σ-+-⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛σ-+⎪⎭⎫ ⎝⎛-=r r r r r S H S r H H K r S H H rK dH V d H H K r S H S H r S H K S H dH dVdV/dH is zero when222σ+=r rK H and, for this value of H , d 2V /dH 2 is negative indicating that it gives the maximum value of V .The value of the perpetual American put is maximized if it is exercised when S equals this value of H . Hence the value of the perpetual American put is 2/2)(σ-⎪⎭⎫ ⎝⎛-r H S H Kwhen H =2rK /(2r +σ2). The value is 2/22222)2(2σ-⎪⎪⎭⎫ ⎝⎛σ+σ+σr rK r S r K This is consistent with the more general result produced in Chapter 26 for the case where the stock provides a dividend yield.Problem 15.24.A company has an issue of executive stock options outstanding. Should dilution be taken into account when the options are valued? Explain you answer.The answer is no. If markets are efficient they have already taken potential dilution intoaccount in determining the stock price. This argument is explained in Business Snapshot 15.3.Problem 15.25.A company’s stock price is $50 and 10 million shares are outstanding. The company is considering giving its employees three million at-the-money five-year call options. Option exercises will be handled by issuing more shares. The stock price volatility is 25%, the five-year risk-free rate is 5% and the company does not pay dividends. Estimate the cost to the company of the employee stock option issue.The Black-Scholes-Merton price of the option is given by setting 050S =, 50K =, 005r =., 025σ=., and 5T =. It is 16.252. From an analysis similar to that in Section 15.10 the cost to the company of the options is 1016252125103⨯.=.+ or about $12.5 per option. The total cost is therefore 3 million times this or $37.5 million. Ifthe market perceives no benefits from the options the stock price will fall by $3.75.Further QuestionsProblem 15.26.If the volatility of a stock is 18% per annum, estimate the standard deviation of the percentage price change in (a) one day, (b) one week, and (c) one month.(a) %13.125218=(b) %50.25218=(c) %20.51218=Problem 15.27.A stock price is currently $50. Assume that the expected return from the stock is 18% per annum and its volatility is 30% per annum. What is the probability distribution for the stock price in two years? Calculate the mean and standard deviation of the distribution. Determine the 95% confidence interval.In this case 050S =, 018=.μ and 030=.σ. The probability distribution of the stock price in two years, T S , is lognormal and is, from equation (15.3), given by:⎥⎦⎤⎢⎣⎡⨯⎪⎭⎫ ⎝⎛-+23.0,2209.018.050ln ~ln 2ϕT Si.e.,)42.0,18.4(~ln 2ϕT SThe mean stock price is from equation (15.4)018203650507167e e .⨯.==.and the standard deviation is from equation (15.5)018503183e .⨯=.95% confidence intervals for ln T S are418196042and 418196042.-.⨯..+.⨯.i.e.,335and 501..These correspond to 95% confidence limits for T S of 335501and e e ..i.e.,2852and 15044..Problem 15.28. (Excel file)Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows:30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0,32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2Estimate the stock price volatility. What is the standard error of your estimate?The calculations are shown in the table below2009471001145i i uu =.=.∑∑and an estimate of standard deviation of weekly returns is:002884=.The volatility per annum is therefore 002079.=. or 20.79%. The standard error of this estimate is 00393=. or 3.9% per annum.Problem 15.29.A financial institution plans to offer a security that pays off a dollar amount equal to 2T S at time T .(a) Use risk-neutral valuation to calculate the price of the security at time t in terms of the stock price, S , at time t . (Hint: The expected value of 2T S can be calculatedfrom the mean and variance of T S given in section 15.1.)(b) Confirm that your price satisfies the differential equation (15.16).(a) The expected value of the security is 2[()]T E S From equations (15.4) and (15.5), at time t :2()2()()2()var()[1]T t T T t T t T E S Se S S e e μμσ---==-Since 22var()[()][()]T T T S E S E S =-, it follows that 22[()]var()[()]T T T E S S E S =+ so that 22222()()22()(2)()2[()][1]T t T t T t T T t E S S e e S e S e μσμμσ---+-=-+=In a risk-neutral world r μ= so that222(2)()ˆ[()]r T t T E S S e σ+-= Using risk-neutral valuation, the value of the derivative security at time t is()2ˆ[()]r T t Te E S --22(2)()()r T t r T t S e e σ+---=22()()r T t S e σ+-=(b) If: 22222()()()()22()()2()()2()22r T t r T t r T t r T t f S e f S r e tf Se Sf e S σσσσσ+-+-+-+-=∂=-+∂∂=∂∂=∂ The left-hand side of the Black-Scholes –Merton differential equation is:222222()()2()()22()()()()2()2r T t r T t r T t r T t S r e rS e S e rS e rfσσσσσσ+-+-+-+--+++==Hence the Black-Scholes equation is satisfied.Problem 15.30.Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months.a. What is the price of the option if it is a European call?b. What is the price of the option if it is an American call?c. What is the price of the option if it is a European put?d. Verify that put –call parity holds.In this case 030S =, 29K =, 005r =., 025=.σ and 412T =/2104225d ==.2202782d ==.(04225)06637(02782)06096N N .=.,.=.(04225)03363(02782)03904N N -.=.,-.=.a. The European call price is00541230066372906096252e -.⨯/⨯.-⨯.=.or $2.52.b. The American call price is the same as the European call price. It is $2.52.c. The European put price is00541229039043003363105e -.⨯/⨯.-⨯.=.or $1.05.d. Put-call parity states that:rT p S c Ke -+=+In this case 252c =., 030S =, 29K =, 105p =. and 09835rT e -=. and it is easy to verify that the relationship is satisfied,Problem 15.31.Assume that the stock in Problem 15.30 is due to go ex-dividend in 1.5 months. The expected dividend is 50 cents.a. What is the price of the option if it is a European call?b. What is the price of the option if it is a European put?c. If the option is an American call, are there any circumstances when it will beexercised early?a. The present value of the dividend must be subtracted from the stock price. This givesa new stock price of:01250053005295031e -.⨯.-.=.and 2103068d ==. 2201625d ==.12()06205()05645N d N d =.;=. The price of the option is therefore005412295031062052905645221e -.⨯/.⨯.-⨯.=.or $2.21.b. Because12()03795()04355N d N d -=.,-=. the value of the option when it is a European put is005412290435529503103795122e -.⨯/⨯.-.⨯.=.。
2013 AMC 10A Problems
2013 AMC 10A ProblemsProblem 1A taxi ride costs $1.50 plus $0.25 per mile traveled. How much does a 5-mile taxi ride cost?（A）2.25 （B）2.50 （C）2.75 （D）3.00 （E）3.75 Problem 2Alice is making a batch of cookies and needs 2.5cups of sugar. Unfortunately, her measuring cup holds only 1/4cup of sugar. How many times must she fill that cup to get the correct amount of sugar?（A）8 （B）10 （C）12 （D）16 （E）20 Problem 3∆ is 40. What Square ABCD has side length 10. Point E is on BC, and the area of ABEis BE?（A）4 （B）5 （C）6 （D）7 （E）8Problem 4A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?（A）35 （B）40 （C）45 （D）50 （E）55 Problem 5Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, Tom gave Sammy dollars, and Dorothy gave Sammy dollars. What isdt-?（A）15 （B）20 （C）25 （D）30 （E）35 Problem 6Joey and his five brothers are ages 3, 5, 7, 9, 11, and 13. One afternoon two of his brothers whose ages sum to 16 went to the movies, two brothers younger than 10 went to play baseball, and Joey and the 5-year-old stayed home. How old is Joey?（A）3 （B）7 （C）9 （D）11 （E）13A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?（A ）6 （B ）8 （C ）9 （D ）12 （E ）16Problem 8 What is the value of 20122014201220142222-+ ? （A ）-1 （B ）1 （C ）35 （D ）2013 （E ）40242 Problem 9In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20% of her three-point shots and 30% of her two-point shots. Shenille attempted 30 shots. How many points did she score?（A ）12 （B ）18 （C ）24 （D ）30 （E ）36Problem 10A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?（A ）15 （B ）30 （C ）40 （D ）60 （E ）70Problem 11A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10 ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?（A ）10 （B ）12 （C ）15 （D ）18 （E ）25Problem 12In △ABC, AB=AC=28 and BC =20. Points D, E, and F are on sides ,,BC AB and ,AC respectively, such that DE and EF are parallel to AC and AB , respectively. What is the perimeter of parallelogram ADEF ?（A ）48 （B ）52 （C ）56 （D ）60 （E ）72How many three-digit numbers are not divisible by 5, have digits that sum to less than 20, and have the first digit equal to the third digit?（A ）52 （B ）60 （C ）66 （D ）68 （E ）70Problem 14A solid cube of side length 1 is removed from each corner of a solid cube of side length 3. How many edges does the remaining solid have?（A ）36 （B ）60 （C ）72 （D ）84 （E ）108Problem 15Two sides of a triangle have lengths 10 and 15. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?（A ）6 （B ）8 （C ）9 （D ）12 （E ）18Problem 16A triangle with vertices (6,5),(8,-3), and (9,1) is reflected about the lineto create asecond triangle. What is the area of the union of the two triangles?（A ）9 （B ）328 （C ）10 （D ）331 （E ）332 Problem 17Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365-day period will exactly two friends visit her?（A ）48 （B ）54 （C ）60 （D ）66 （E ）72Problem 18Let points A =(0,0), B =(1,2), C =(3,3), and D =(4,0). Quadrilateral ABCD is cut into equal area pieces by a line passing through A . This line intersects CD at point ),(sr q p , where these fractions are in lowest terms. What is s r q p +++ ?（A ）54 （B ）58 （C ）62 （D ）70 （E ）75Problem 19In base 10, the number 2013 ends in the digit 3. In base 9, on the other hand, the same number is written as (2676)9 and ends in the digit 6. For how many positiveintegers does the base--representation of 2013 end in the digit 3?（A ）6 （B ）9 （C ）13 （D ）16 （E ）18A unit square is rotated 45º about its center. What is the area of the region swept out by the interior of the square?（A ）4221π+- （B ）421π+ （C ）422π+- （D ）422π+（E ）8421π++ Problem 21A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The th k pirate to take a share takes k/12 of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the th 12 pirate receive?（A ）720 （B ）1296 （C ）1728 （D ）1925 （E ）3850 Problem 22Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?（A ）2 （B ）23 （C ）35 （D ）3 （E ）2 Problem 23In △ABC, AB =86, and AC =97. A circle with center A and radius AB intersects BC at points B and X . Moreover BX and CX have integer lengths. What is BC ?（A ）11 （B ）28 （C ）33 （D ）61 （E ）72 Problem 24Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?（A ）540 （B ）600 （C ）720 （D ）810 （E ）900 Problem 25All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?（A ）49 （B ）65 （C ）70 （D ）96 （E ）1282013 AMC 10A SolutionsProblem 1There are five miles which need to be traveled. The cost of these five milesis (0.25*5)=1.25. Adding this to 1.5, we get 2.75. Problem 2To get how many cups we need, we realize that we simply need to divide the number of cups needed by the number of cups collected in her measuring cup each time. Thus, weneed to evaluate the fraction 41212. Simplifying, this is equal to 10)4(25= Problem 3We know that the area of ABE ∆ is equal to 2)(BE AB . Plugging in AB =10, we get80=10BE. Dividing, we find that BE=8. Problem 4We know that, for the games where they scored an odd number of runs, they cannot have scored twice as many runs as their opponents, as odd numbers are not divisible by 2. Thus, from this, we know that the five games where they lost by one run were when they scored 1, 3, 5, 7, and 9 runs, and the others are where they scored twice as many runs. We can make the following chart:The sum of their opponent's scores is 2+1+4+2+6+3+8+4+10+5=45 Problem 5Solution 1The total amount paid is 105+125+175=405. To get how much each should have paid, we do 405/3=135.Thus, we know that Tom needs to give Sammy 30 dollars, and Dorothy 10 dollars. Thismeans that 201030=-=-d t . Solution 2The difference in the money that Tommy paid and Dorothy paid is 20. In order for themboth to have paid the same amount, Tommy must pay 20 more than Dorothy.Problem 6Because the 5-year-old stayed home, we know that the 11-year-old did not go to the movies, as the 5-year-old did not and 11+5=16. Also, the 11-year-old could not have gone to play baseball, as he is older than 10. Thus, the 11-year-old must have stayed home, soProblem 7Solution 1Let us split this up into two cases.Case 1: The student chooses both algebra and geometry.This means that 3 courses have already been chosen. We have 3 more options for the last course, so there are 3 possibilities here.Case 2: The student chooses one or the other.Here, we simply count how many ways we can do one, multiply by 2, and then add to the previous.WLOG assume the mathematics course is algebra. This means that we can choose 2 of History, Art, and Latin, which is simply . If it is geometry, we have another 3 options, so we have a total of 6 options if only one mathematics course is chosen.waysSolution 2We can use complementary counting. Since there must be an English class, we will add that to our list of classes for 3 remaining spots for the classes. We are also told that there needs to be at least one math class. This calls for complementary counting. The totalnumber of ways of choosing 3 classes out of the 5 is . The total number of ways ofchoosing only non-mathematical classes is . Therefore the amount of ways you canpick classes with at least one math class is ways.Problem 8Factoring out, we get: )12(2)12(22201222012-+ .we see that it simplifies to 5/3. Problem 9Let the number of attempted three-point shots made be and the number of attemptedtwo-point shots be y . We know that 30=+y x , and we need to evaluate y x )23.0()32.0(∙+∙, as we know that the three-point shots are worth 3 points and that she made 20% of them and that the two-point shots are worth 2 and that she made 30%of them.Simplifying, we see that this is equal to )(6.06.06.0y x y x +=+. Plugging in 30=+y x ,we get 0.6(30)=18. Problem 10Let the total amount of flowers be x . Thus, the number of pink flowers is x 6.0, and the number of red flowers is x 4.0. The number of pink carnations isx x 4.0)6.0(32= and the number of red carnations is x x 3.0)4.0(43=. Summing these, the total number ofcarnations is x x x 7.03.04.0=+. Dividing, we see that %707.0/7.0==x x Problem 11Let the number of students on the council be . We know that there are ways tochoose a two person team. This gives that 20)1(=-x x , which has a positive integer solution of 5.If there are 5 people on the welcoming committee, then there areways tochoose a three-person committee. Problem 12Note that because DE and EF are parallel to the sides of △ABC , the internal triangles △BDE and △EFC are similar to △ABC , and are therefore also isosceles triangles. It follows that BD=DE . Thus, AD+DE=AD+DB=AB =28.Since opposite sides of parallelograms are equal, the perimeter is 2* (AD+DE Problem 13These three digit numbers are of the form xyx . We see that 0≠x and 5≠x , as 0=x does not yield a three-digit integer and 5=x yields a number divisible by 5.The second condition is that the sum 202<+y x . When is 1,2,3, or 4, y can be anydigit from 0 to 9, as 102<x . This yields 10(4)=40 numbers.When 6=x , we see that 2012<+y so 8<y . This yields 8 more numbers.When 7=x , 2014<+y so 6<y . This yields 6 more numbers.When 8=x , 2016<+y so 4<y . This yields 4 more numbers.When 9=x , 2018<+y so 2<y . This yields 2 more numbers.Summing, we get 40+8+6+4+2=60. Problem 14We can use Euler's polyhedron formula that says that F+V=E +2. We know that there are originally 6 faces on the cube, and each corner cube creates 3 more. 6+8(3)=30. In addition, each cube creates 7 new vertices while taking away the original 8, yielding8(7)=56 vertices. Thus E +2=56+30, so E =84. Problem 15Solution 1 (Meta)The shortest side length has the longest altitude perpendicular to it. The average of the two altitudes given will be between the lengths of the two altitudes, therefore the length of the side perpendicular to that altitude will be between 10 and 15. The only answer choicethat meets this requirement is 12. Solution 2Let the height to the side of length 15 be 1h , the height to the side of length 10 be 2h , the area be A , and the height to the unknown side be 3h .Because the area of a triangle is 2/bh , we get that A h 2)(151= and A h 2)(102= , so, setting them equal, 2/312h h =. From the problem, we know that 2132h h h +=. Substituting, we get that 1325.1h h =. Thus, the side length is going to be121525.12451==h A . Problem 16Let A be at (6,5), B be at (8,-3), and C be at (9,1). Reflecting over the line 8=x , we see that A ’=D =(10,5), B ’=B (as the x-coordinate of B is 8), and C ’=E =(7,1). Line AB can be represented as 294+-=x y , so we see that E is on line AB .We see that if we connect A to D , we get a line of length 4 (between (6,5) and (10,5)). The area of ABD ∆ is equal to 162/)8(42/==bh .Now, let the point of intersection between AC and DE be F . If we can just find the area of △ADF and subtract it from 16, we are done.We realize that because the diagram is symmetric over 8=x , the intersection of lines AC and DE should intersect at an x-coordinate of 8. We know that the slope of DE is (5-1)/(10-7)=4/3. Thus, we can represent the line going through E and D as)7(341-=-x y . Plugging in 8=x , we find that the y-coordinate of F is 7/3. Thus, the height of ADF ∆ is 38375=-. Using the formula for the area of a triangle, the area of ADF ∆ is 16/3.To get our final answer, we must subtract this from 16. [][]3/323/1616=-=-ADF ABD Problem 17The 365-day time period can be split up into 6 60-day time periods, because after 60 days, all three of them visit again (Least common multiple of 3, 4, and 5). You can find how many times each pair of visitors can meet by finding the LCM of their visiting days and dividing that number by 60. Remember to subtract 1, because you do not wish to count the 60th day, when all three visit.A andB visit 414*360=- times. B andC visit 215*460=- times. C and A visit 315*360=- times. This is a total of 9 visits per 60 day period. Therefore, the total number of 2-person visits is9*6=54. Problem 18First, we shall find the area of quadrilateral ABCD . This can be done in any of three ways: Pick's Theorem: []215127512=-+=-+=B I ABCD Splitting: Drop perpendiculars from B and C to the x-axis to divide the quadrilateral into triangles and trapezoids, and so the area is 1+5+3/2=15/2.Shoelace Theorem: The area is half of 15433231=∙-∙-∙ , or 15/2. []2/15=ABCD . Therefore, each equal piece that the line separates ABCD into must have an area of 15/4.Call the point where the line through A intersects CD E . We know that []2/4/15bh ADE ==. Furthermore, we know that ,4=b as 4=AD . Thus, solving for , we find that 4/152=h , so 8/15=h . This gives that the y coordinate of E is 15/8.Line CD can be expressed as 123+-=x y , so the coordinate of E satisfies1238/15+-=x . Solving for , we find that 8/27=x .From this, we know that E=(27/8, 15/8). 27+15+8+8=58.Problem 19We want the integers such that b b ⇒≡)(mod 32013 is a factor of 2010. Since2010=2·3·5· 67 it has (1+1) (1+1) (1+1) (1+1)=16 factors. Since cannot equal1,2,or 3,Problem 20Solution 1First, we need to see what this looks like. Below is a diagram.For this square with side length 1, the distance from center to vertex is 2/1=r , hence the area is composed of a semicircle of radius , plus 4 times a parallelogram with height1/2 and base)21(22+. That is to say, the total area is 224)21(424)2/1(212-+=++ππ.Solution 2Let O be the center of the square and C be the intersection of OB and AD . The desired area consists of the unit square, plus 4 regions congruent to the region bounded byarc ,,AC AB and BC , plus 4 triangular regions congruent to right triangle BCD . The areaof the region bounded by arc ,,AC AB and BC is. Since the circle has radius 2/1 , the area of the region is 812-π, so 4 times the area of that region is 214-π. Now we find the area of △BCD . 2122-=-=OC BO BC . Since △BCD is a 45—45—90 right triangle, the area of △BCD is 2)(2221222-=BC , so 4 times the area of △BCD is 223-. Finally, the area of the whole region is 224)214()223(1-+=-+-+ππ. Problem 21Let be the number of coins. After thepirate takes his share,1212k -of the original amount is left. Thus, we know thatmust be an integer. Simplifying, we get. Now, the minimal is the denominator of thisfraction multiplied out, obviously. We mentioned before that this product must be an integer. Specifically, it is an integer and it is the amount that the th 12 pirate receives, as he receives 12/12=1= all of what is remaining.Thus, we know the denominator is cancelled out, so the number of gold coins received isgoing to be the product of the numerators, 11·5·7·5=1925. Problem 22Solution 1Set up an isosceles triangle between the center of the 8th sphere and two opposite ends of the hexagon. Then set up another triangle between the point of tangency of the 7th and 8th spheres, and the points of tangency between the 7th sphere and 2 of the original spheres on opposite sides of the hexagon. Express each side length of the triangles in terms of r (the radius of sphere 8) and h (the height of the first triangle). You can then use Pythagorean Theorem to set up two equations for the two triangles, and find the values of h and r.2222)1(h r +=+ 222)(3)23(r h ++= 2/3=r Solution 2We have a regular hexagon with side length 2 and six spheres on each vertex with radius 1 that are internally tangent, therefore, drawing radii to the tangent points would create this regular hexagon.Imagine a 2D overhead view. There is a larger sphere which the 6 spheres are internally tangent to, with center in the center of the hexagon. To find the radius of the larger sphere we must first, either by prior knowledge or by deducing from the angle sum that the hexagon can be split into 6 equilateral triangles from its vertices, that the radius is two more than the small radius, or 3.Now imagine the figure in three dimensions. The 6th sphere must be sitting atop of the 6 spheres, which is the only possibility for it to be tangent to all the 6 small spheresexternally and the larger sphere internally. The ring of small spheres is symmetrical and the 8th sphere will be resting atop it with its center aligned with the diameter of the large sphere.We can now create a right triangle with one leg being the line from a vertex of the hexagon to the center of the hexagon and one leg being the line from the center of the 7th sphere to the center of the 8th sphere. Let the radius of our 8th sphere be . As previouslymentioned, the distance from the center of the hexagon to one of its vertices is 2. The distance between the centers will be 3-r . The hypotenuse will be 1+r .We now have a right triangle. Applying the Pythagorean Theorem, 222)1()3(2r r +=-+. Expanding and solving for , we find r =12/8=3/2.Problem 23Solution 1 (Power of a Point)Let ,,p CX q BX == and AC meet the circle at Y and Z , with Y on AC. Then AZ=AY =86.Using the Power of a Point, we get that 61*3*11)183(11)(==+q p p . We know that p q p >+, and that 13>p by the triangle inequality on △ACX . Thus, we get that61=+=q p BC Solution 2 (Stewart's Theorem)Let represent CX , and let represent BX . Since the circle goesthrough B and X, AB=AX=86. Then by Stewart's Theorem,22222297868697)(86)(=++⇒+=+++xy x x y y x y x xy(Since cannot be equal to 0, dividing both sides of the equation by is allowed.)2013)()8697)(8697()(=+⇒-+=+y x x y x xThe prime factors of 2013 are 3, 11, and 61. Obviously, y x x +< . In addition, by the Triangle Inequality, BC<AB+AC , so 183<+y x . Therefore, must equal 33, andy x + must equal 61. Problem 24Let us label the players of the first team A, B, and C , and those of the second team, X, Y , and Z .1. One way of scheduling all six distinct rounds could be: Round 1---->Round 2---->Round 3---->Round 4---->Round 5---->Round 6---->The above mentioned schedule ensures that each player of one team plays twice with each player from another team. Now you can generate a completely new schedule bypermuting those 6 rounds and that can be done in =ways. 2. One can also make the schedule in such a way that two rounds are repeated. (a) Round 1---->Round 2---->Round 3---->Round 4---->Round 5---->Round 6----> (b)Round 1---->Round 2---->Round 3---->Round 4---->Round 5---->Round 6---->As mentioned earlier any permutation of (a) and (b) will also give us a new schedule. For both (a) and (b) the number of permutations are 90!2!2!2!6So the total number of schedules is 720+90+90=900. Problem 25Solution 1 (Drawing)If you draw a good diagram like the one below, it is easy to see that there are 49Solution 2 (Elimination)Let the number of intersections be . We know that , as every 4 points formsa quadrilateral with intersecting diagonals. However, four diagonals intersect in the center,so we need to subtract from this count. 70-5=65. Note that diagonals like ,,CG AD and BE all intersect at the same point. There are 8 of this type with threediagonals intersecting at the same point, so we need to subtract 2 of the(one is keptas the actual intersection). In the end, we obtain 65-16=49.2013 AMC 10A Answer Key1. C2. B3.E4. C5. B6.D7.C8.C 8. B 10.E11.A 12. C 13.B 14.D 15. D 16.E 17. B 18.B 19.C 20.C21.D 22.B 23.D 24.E 25.A。
2017-2020AMC真题和答案
2020 AMC 12A Problems Problem 1Carlos took of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?Problem 2The acronym AMC is shown in the rectangular grid below with grid linesspaced unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMCProblem 3A driver travels for hours at miles per hour, during which her cargets miles per gallon of gasoline. She is paid per mile, and her only expense is gasoline at per gallon. What is her net rate of pay, in dollars per hour, after this expense?Problem 4How many -digit positive integers (that is, integers between and , inclusive) having only even digits are divisible byProblem 5The integers from to inclusive, can be arranged to form a -by-square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?Problem 6In the plane figure shown below, of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetryProblem 7Seven cubes, whose volumes are , , , , , , and cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?Problem 8What is the median of the following list of numbersProblem 9How many solutions does the equation have on the intervalProblem 10There is a unique positive integer suchthat What is the sum of the digitsofProblem 11A frog sitting at the point begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length , and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square withvertices and . What is the probability that thesequence of jumps ends on a vertical side of the squareProblem 12Line in the coordinate plane has the equation . Thisline is rotated counterclockwise about the point to obtain line . What is the -coordinate of the -intercept of lineProblem 13There are integers , , and , each greater than 1, suchthat for all . What is ?Problem 14Regular octagon has area . Let be the area ofquadrilateral . What isProblem 15In the complex plane, let be the set of solutions to andlet be the set of solutions to . What is the greatest distance between a point of and a point ofProblem 16A point is chosen at random within the square in the coordinate plane whose vertices are and . Theprobability that the point is within units of a lattice point is . (A point is a lattice point if and are both integers.) What is to the nearest tenthProblem 17The vertices of a quadrilateral lie on the graph of , and the -coordinates of these vertices are consecutive positive integers. The area of thequadrilateral is . What is the -coordinate of the leftmost vertex?Problem 18Quadrilateral satisfies, and . Diagonals and intersect at point , and . What is the area of quadrilateral ?Problem 19There exists a unique strictly increasing sequence of nonnegativeintegers suchthat What isProblem 20Let be the triangle in the coordinate plane with vertices , ,and . Consider the following five isometries (rigid transformations) of the plane: rotations of , , and counterclockwise around the origin, reflection across the -axis, and reflection across the -axis. How many ofthe sequences of three of these transformations (not necessarily distinct) will return to its original position? (For example, a rotation, followed by a reflection across the -axis, followed by a reflection across the -axis will return to its original position, but a rotation, followed by a reflection across the -axis, followed by another reflection across the -axis will not return to its original position.)Problem 21How many positive integers are there such that is a multiple of , and the least common multiple of and equals times the greatest common divisor of andProblem 22Let and be the sequences of real numbers suchthat for all integers , where . What isProblem 23Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly . Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?Problem 24Suppose that is an equilateral triangle of side length , with the property that there is a unique point inside the triangle suchthat , , and . What isProblem 25The number , where and are relatively prime positive integers, has the property that the sum of all realnumbers satisfying is , where denotes thegreatest integer less than or equal to and denotes the fractional part of . What is2020 AMC 12A Answer Key1. C2. C3. E4. B5. C6. D7. B8. C9. E10.E11.B12.B13.B14.B15.D16.B17.D18.D19.C20.A21.D22.B23.A24.B25.C2020 AMC 12B Problems Problem 1What is the value in simplest form of the followingexpression?Problem 2What is the value of the followingexpression?Problem 3The ratio of to is , the ratio of to is , and the ratioof to is . What is the ratio of to ?Problem 4The acute angles of a right triangle are and , where andboth and are prime numbers. What is the least possible value of ?Problem 5Teams and are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team has won of itsgames and team has won of its games. Also, team has won moregames and lost more games than team How many games hasteam played?Problem 6For all integers the value of is always whichof the following?Problem 7Two nonhorizontal, non vertical lines in the -coordinate plane intersect to form a angle. One line has slope equal to times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines?Problem 8How many ordered pairs of integers satisfy theequationProblem 9A three-quarter sector of a circle of radius inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by tapingtogether along the two radii shown. What is the volume of the cone in cubicinches?Problem 10In unit square the inscribedcircle intersects at and intersects at a point different from What isProblem 11As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?Problem 12Let be a diameter in a circle of radius Let be a chord in the circle that intersects at a point suchthat and What isProblem 13Which of the following is the value ofProblem 14Bela and Jenn play the following game on the closed interval of the real number line, where is a fixed integer greater than . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in theinterval . Thereafter, the player whose turn it is chooses a real numberthat is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?Problem 15There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?Problem 16An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?How many polynomials of theform , where , , ,and are real numbers, have the property that whenever is a root, sois ? (Note that )Problem 18In square , points and lie on and , respectively, so that Points and lie on and , respectively, and points and lie on so that and . See the figure below. Triangle , quadrilateral ,quadrilateral , and pentagon each has area Whatis ?Square in the coordinate plane has vertices at thepoints and Consider the following four transformations: a rotation of counterclockwise around the origin; a rotation of clockwise around the origin; a reflection across the -axis; and a reflection across the -axis.Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying and then wouldsend the vertex at to and would send thevertex at to itself. How many sequences of transformations chosen from will send all of the labeled vertices back to theiroriginal positions? (For example, is one sequenceof transformations that will send the vertices back to their original positions.)Problem 20Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?Problem 21How many positive integers satisfy(Recallthat is the greatest integer not exceeding .)Problem 22What is the maximum value of for real values ofProblem 23How many integers are there such that whenever are complex numbers such thatthen the numbers are equally spaced on the unit circle in the complex plane?Problem 24Let denote the number of ways of writing the positive integer as a product where , the are integers strictly greater than , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct).For example, the number can be written as , , and ,so . What is ?Problem 25For each real number with , let numbers and be chosen independently at random from the intervals and , respectively, and let be the probability thatWhat is the maximum value of2020 AMC 12B Answer Key1. C2. A3. E4. D5. C6. D7. C8. D9. C10. B11. D12. E13. D14. A15. C16. B17. C18. B19. C20. D21. C22. C23. B24. A25. BThe area of a pizza with radius is percent larger than the area of a pizza with radius inches. What is the integer closest to ?Solution Supposeis of . What percent of is ?Solution A box containsred balls,green balls, yellow balls,blue balls, white balls, and black balls. What is the minimum numberof balls that must be drawn from the box without replacement to guarantee that at least balls of a single color will be drawn?SolutionWhat is the greatest number of consecutive integers whose sum is ?SolutionTwo lines with slopes andintersect at . What is the area of the triangle enclosed by these two lines and the line?SolutionThe figure below shows line with a regular, infinite, recurring pattern of squares and line segments.How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?some rotation around a point of linesome translation in the direction parallel to linethe reflection across linesome reflection across a line perpendicular to lineSolutionMelanie computes the mean , the median, and the modes of the values that are the dates in the months of . Thus her data consist of , , . . . , , , , and . Let be the median of the modes. Which of the following statements is true?SolutionProblem 2Problem 3Problem 4Problem 5Problem 6Problem 7For a set of four distinct lines in a plane, there are exactly distinct points that lie on two or more of the lines. What is the sum of allpossible values of ?Solution A sequence of numbers is defined recursively by, , and for all . Then can be written as , whereandare relatively prime positive inegers. What is Solution The figure below shows circles of radius within a larger circle. All the intersections occur at points of tangency. What is the area ofthe region, shaded in the figure, inside the larger circle but outside all the circles of radius?Solution For some positive integer , the repeating base- representation of the (base-ten) fraction is. What is ?Solution Positive real numbers and satisfy and. What is ?Problem 8Problem 9Problem 10Problem 11Problem 12Solution How many ways are there to paint each of the integers either red, green, or blue so that each number has a different colorfrom each of its proper divisors?Solution For a certain complex number, the polynomial has exactly 4 distinct roots. What is?Solution Positive real numbers andhave the property that and all four terms on the left are positive integers, where denotes the base- logarithm. What is?Solution The numbers are randomly placed into the squares of a grid. Each square gets one number, and each of thenumbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?Solution Let denote the sum of the th powers of the roots of the polynomial . In particular, ,, and . Let , , and be real numbers such that for ,, What is?Solution A sphere with center has radius . A triangle with sides of length and is situated in space so that each of its sides is tangent to the sphere. What is the distance between and the plane determined by the triangle?SolutionIn with integer side lengths,What is the least possible perimeter for ?Problem 13Problem 14Problem 15Problem 16Problem 17Problem 18Problem 19Real numbers between and , inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is if the second flip is heads and if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval . Two random numbersandare chosen independently in thismanner. What is the probability that?Solution Let What is Solution Circles and , both centered at , have radii and , respectively. Equilateral triangle , whose interior lies in the interior of but in the exterior of , has vertex on , and the line containing side is tangent to . Segments and intersect at ,and . Thencan be written in the form for positive integers ,, , with . What is ?Solution Define binary operations andby for all real numbers and for which these expressions are defined. The sequence is defined recursively byand for all integers . To the nearest integer, what is ?SolutionFor how many integers between and , inclusive, isan integer? (Recall that .)Problem 21Problem 22Problem 23Problem 24Copyright © 2019 Art of Problem SolvingLet be a triangle whose angle measures are exactly , , and . For each positive integer define to be the foot of the altitude from to line . Likewise, define to be the foot of the altitude from to line, and to be the foot of the altitude from to line. What is the least positive integer for whichis obtuse?Solution2019 AMC 12A (Problems • Answer Key • Resources (/Forum/resources.php?c=182&cid=44&year=2019))Preceded by2018 AMC 12B Problems Followed by 2019 AMC 12B Problems1 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18• 19 • 20 • 21 • 22 • 23 • 24 • 25All AMC 12 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Retrieved from "https:///wiki/index.php?title=2019_AMC_12A_Problems&oldid=101818"See also2019 AMC 12A Answer Key1. E2. D3. B4. D5. C6. C7. E8. D9. E10. A11. D12. B13. E14. E15. D16. B17. D18. D19. A20. B21. C22. E23. D24. D25. E2019 AMC 12B Answer Key1. D2. E3. E4. C5. B6. A7. A8. A9. B10. E11. D12. D13. C14. C15. E16. A17. D18. C19. B20. C21. B22. C23. C24. C25. C2018 AMC 12A ProblemsA large urn contains balls, of which are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be ? (No red balls are to be removed.)SolutionWhile exploring a cave, Carl comes across a collection of -pound rocks worth each, -pound rocks worth each, and -pound rocks worth each. There are at least of each size. He can carry at most pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?SolutionSolutionAlice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Letbe the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?Solution What is the sum of all possible values of for which the polynomials andhave a root in common?Solution For positive integers andsuch that , both the mean and the median of the set are equalto . What is ?SolutionFor how many (not necessarily positive) integer values of is the value of an integer?SolutionAll of the triangles in the diagram below are similar to iscoceles triangle , in which . Each of the 7 smallest triangles has area 1, andhas area 40. What is the area of trapezoid ?Solution Which of the following describes the largest subset of values ofwithin the closed interval for which for everybetweenand , inclusive?Solution How many ordered pairs of real numbers satisfy the following system of equations?SolutionA paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?Problem 5Problem 6Problem 7Problem 8Problem 9Problem 10Problem 11Solution Let be a set of 6 integers taken fromwith the property that ifand are elements of with , thenis not a multiple of . What is the least possible value of an element in Solution How many nonnegative integers can be written in the form where for ?Solution The solutions to the equation, where is a positive real number other than or , can be written as where and are relatively primepositive integers. What is ?Solution A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the totalnumber of possible symmetric scanning codes?Solution Which of the following describes the set of values of for which the curvesand in the real -plane intersect at exactly points?SolutionFarmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?Problem 12Problem 13Problem 14Problem 15Problem 16Problem 17SolutionTriangle with and has area . Let be the midpoint of, and let be the midpoint of . The angle bisector of intersects andat and , respectively. What is the area of quadrilateral ?Solution Letbe the set of positive integers that have no prime factors other than , , or . The infinite sum of the reciprocals of the elements of can be expressed as , where and are relatively prime positive integers. What is ?Solution Triangle is an isosceles right triangle with . Let be the midpoint of hypotenuse . Points and lie on sidesand ,respectively, so that andis a cyclic quadrilateral. Given that triangle has area , the length can be written as , where ,, and are positive integers andis not divisible by the square of any prime. What is the value of?Solution Which of the following polynomials has the greatest real root?Solution The solutions to the equations and where form the vertices of a parallelogram in the complex plane. The areaof this parallelogram can be written in the formwhereand are positive integers and neither nor is divisible by the square of any prime number. What is Solution In and Points andlie on sides and respectively, so that Let andbe the midpoints of segments and respectively. What is the degree measure of the acute angle formed by lines and Solution Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between and Armed with this information, what number should Carol choose to maximize her chance of winning? Solution For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the-digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value offorwhich there are at least two values of such that ?SolutionProblem 18Problem 19Problem 20Problem 21Problem 22Problem 23Problem 24Problem 25See also2018 AMC 12A (Problems • Answer Key • Resources)Preceded by 2017 AMC 12B ProblemsFollowed by 2018 AMC 12B Problems1 ·2 ·3 ·4 ·5 ·6 ·7 ·8 ·9 · 10 · 11 · 12 · 13 · 14 · 15 · 16 · 17 · 18 · 19 · 20 · 21 · 22 · 23 · 24 · 25All AMC 12 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.Retrieved from "/wiki/index.php?title=2018_AMC_12A_Problems&oldid=94197"Category: AMC 12 ProblemsCopyright © 2018 Art of Problem Solving2018 AMC 12A Answer Key1. D2. C3. E4. D5. E6. B7. E8. E9. E10. C11. D12. C13. D14. D15. B16. E17. D18. D19. C20. D21. B22. A23. E24. B25. DRetrieved from "/wiki/index.php?title=2018_AMC_12A_Answer_Key&oldid=90552"Copyright © 2018 Art of Problem Solving2018 AMC 12B ProblemsKate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches.How many pieces of cornbread does the pan contain?SolutionSam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30minutes?Solution A line with slope 2 intersects a line with slope 6 at the point . What is the distance between the -intercepts ofthese two lines? Solution A circle has a chord of length , and the distance from the center of the circle to the chord is . What is the area of thecircle?Solution How many subsets ofcontain at least one prime number?Solution Suppose cans of soda can be purchased from a vending machine for quarters. Which of the following expressionsdescribes the number of cans of soda that can be purchased fordollars, where 1 dollar is worth 4 quarters?SolutionWhat is the value ofSolutionProblem 2Problem 3Problem 4Problem 5Problem 6Problem 7Problem 8Line segment is a diameter of a circle with . Point, not equal toor, lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?SolutionWhat isSolution A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?SolutionA closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figureon the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length and height . What is the area of the sheet of wrapping paper?Solution Side of has length . The bisector of angle meets at , and . The set of all possiblevalues of is an open interval . What is ?Problem 9Problem 10Problem 11Problem 12Solution Square has side length . Point lies inside the square so thatand . The centroids of, ,, and are the vertices of a convex quadrilateral. What is the area of that quadrilateral?Solution Joey, Chloe, and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?SolutionHow many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3?Solution The solutions to the equation are connected in the complex plane to form a convex regular polygon,three of whose vertices are labeled and . What is the least possible area ofSolutionLet and be positive integers such thatand is as small as possible. What is ?Problem 13Problem 14Problem 15Problem 16Problem 17。
chapter 9投资作业-chapter 15习题
习题Chapter 93. You are a consultant to a large manufacturing corporation that is considering a project with the following net after-tax cash flows (in millions of dollars):The project’s beta is 1.8. Assuming that, what is the net present value of the project? What is the highest possible beta estimate for the project before its NPV becomes negative?4. Are the following true or false? Explain.a. Stocks with a beta of zero offer an expected rate of return of zero.b. The CAPM implies that investors require a higher return to hold highly volatile securities.c. You can construct a portfolio with beta of .75 by investing .75 of the investment budget in T-bills and the remainder in the market portfolio.In problems 13 to 15 assume that the risk-free rate of interest is 6% and the expected rate of return on the market is 16%.13. Ashare of stock sells for $50 today. It will pay a dividend of $6 per share at the end of the year. Its beta is 1.2. What do investors expect the stock to sell for at the end of the year?14. I am buying a firm with an expected perpetual cash flow of $1,000 but am unsure of its risk. If I think the beta of the firm is .5, when in fact the beta is really 1, how much more will I offer for the firm than it is truly worth?15. A stock has an expected rate of return of 4%. What is its beta?17. Suppose the rate of return on short-term government securities (perceived to be riskfree) is about 5%. Suppose also that the expected rate of return required by the market for a portfolio with a beta of 1 is 12%. According to the capital asset pricing model (security market line):a. What is the expected rate of return on the market portfolio?b. What would be the expected rate of return on a stock with beta＝0?c. Suppose you consider buying a share of stock at $40. The stock is expected to pay $3 dividends next year and you expect it to sell then for $41. The stock risk has been evaluated at beta＝－0.5. Is the stock overpriced or underpriced?21. The security market line depicts:a. A security’s expected return as a function of its systematic risk.b. The market portfolio as the optimal portfolio of risky securities.c. The relationship between a security’s return and the return on an index.d. The complete portfolio as a combination of the market portfolio and the risk-free asset.22. Within the context of the capital asset pricing model (CAPM), assume:• Expected return on the market ＝15%.• Risk-free rate _ 8%.• Expected rate of return on XYZ security ＝17%.• Beta of XYZ security ＝1.25.Which one of the following is correct?a. XYZ is overpriced.b. XYZ is fairly priced.c. XYZ’s alpha is＝-0.25%.d. XYZ’s alpha is=0 .25%.The following table shows risk and return measures for two portfolios. answer question 26 and 2726. When plotting portfolio R on the preceding table relative to the SML, portfolio R lies:a. On the SML.b. Below the SML.c. Above the SML.d. Insufficient data given.27. When plotting portfolio R relative to the capital market line, portfolio R lies:a. On the CML.b. Below the CML.c. Above the CML.d. Insufficient data given.31. Karen Kay, a portfolio manager at Collins Asset Management, is using the capital asset pricing model for making recommendations to her clients. Her research department has developed the information shown in the following exhibit.a. Calculate expected return and alpha for each stock.b. Identify and justify which stock would be more appropriate for an investor who wants toi. add this stock to a well-diversified equity portfolio.ii. hold this stock as a single-stock portfolio.Chapter 111. A portfolio management organization analyzes 60 stocks and constructs a meanvariance efficient portfolio using only these 60 securities.a. How many estimates of expected returns, variances, and covariances are needed to optimize this portfolio?b. If one could safely assume that stock market returns closely resemble a single index structure, how many estimates would be needed?2. The following are estimates for two of the stocks in problem 1.The market index has a standard deviation of 22% and the risk-free rate is 8%.a. What is the standard deviation of stocks A and B?b. Suppose that we were to construct a portfolio with proportions:Stock A: .30Stock B: .45T-bills: .25Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of the portfolio.4. Consider the two (excess return) index model regression results for A and B:a. Which stock has more firm-specific risk?b. Which has greater market risk?c. For which stock does market movement explain a greater fraction of return variability?d. Which stock had an average return in excess of that predicted by the CAPM?e. If rf were constant at 6% and the regression had been run using total rather than excess returns, what would have been the regression intercept for stock A ?Use the following data for problems 5 through 9. Suppose that the index model for stocks A and B is estimated from excess returns with the following results:5. What is the standard deviation of each stock?6. Break down the variance of each stock to the systematic and firm-specific components.7. What are the covariance and correlation coefficient between the two stocks?8. What is the covariance between each stock and the market index?9. Are the intercepts of the two regressions consistent with the CAPM? Interpret their values.15. Based on current dividend yields and expected growth rates, the expected rates of return on stocks A and B are 11% and 14%, respectively. The beta of stock A is .8, while that of stock B is 1.5. The T-bill rate is currently 6%, while the expected rate of return on the S&P 500 index is 12%. The standard deviation of stock A is 10% annually, while that of stock B is 11%.a. If you currently hold a well-diversified portfolio, would you choose to add either of these stocks to your holdings?b. If instead you could invest only in bills and one of these stocks, which stock would you choose? Explain your answer using either a graph or a quantitative measure of the attractiveness of the stocks.16. Assume the correlation coefficient between Baker Fund and the S&P 500 Stock Index is .70. What percentage of B aker Fund’s total risk is specific (i.e., nonsystematic)?a. 35%b. 49%c. 51%d. 70%17. The correlation between the Charlottesville International Fund and the EAFE Market Index is 1.0. The expected return on the EAFE Index is 11%, the expected return on Charlottesville International Fund is 9%, and the risk-free return in EAFE countries is 3%. Based on this analysis, the implied beta of Charlottesville International is:a. Negativeb. .75c. .82d. 1.00chapter 122. Which of the following most appears to contradict the proposition that the stock market is weakly efficient? Explain.a. Over 25% of mutual funds outperform the market on average.b. Insiders earn abnormal trading profits.c. Every January, the stock market earns abnormal returns.3. Suppose that, after conducting an analysis of past stock prices, you come up with the following observations. Which would appear to contradict the weak form of the efficient market hypothesis? Explain.a. The average rate of return is significantly greater than zero.b. The correlation between the return during a given week and the return during the following week is zero.c. One could have made superior returns by buying stock after a 10% rise in price and selling after a 10% fall.d. One could have made higher-than-average capital gains by holding stocks with low dividend yields.4. Which of the following statements are true if the efficient market hypothesis holds?a. It implies that future events can be forecast with perfect accuracy.b. It implies that prices reflect all available information.c. It implies that security prices change for no discernible reason.d. It implies that prices do not fluctuate.5. Which of the following observations would provide evidence against the semistrong form of the efficient market theory? Explain.a. Mutual fund managers do not on average make superior returns.b. You cannot make superior profits by buying (or selling) stocks after the announcement of an abnormal rise in dividends.c. Low P/E stocks tend to have positive abnormal returns.d. In any year approximately 50% of pension funds outperform the market.6. The semistrong form of the efficient market hypothesis asserts that stock prices:a. Fully reflect all historical price information.b. Fully reflect all publicly available information.c. Fully reflect all relevant information including insider information.d. May be predictable.7. Assume that a company announces an unexpectedly large cash dividend to its shareholders.In an efficient market without information leakage, one might expect:a. An abnormal price change at the announcement.b. An abnormal price increase before the announcement.c. An abnormal price decrease after the announcement.d. No abnormal price change before or after the announcement.8. Which one of the following would provide evidence against the semistrong form of the efficient market theory?a. About 50% of pension funds outperform the market in any year.b. All investors have learned to exploit signals about future performance.c. Trend analysis is worthless in determining stock prices.d. Low P/E stocks tend to have positive abnormal returns over the long run. Chapter 141. Which security has a higher effective annual interest rate?a. A 3-month T-bill selling at $97,645 with par value $100,000.b. A coupon bond selling at par and paying a 10% coupon semiannually.2. Treasury bonds paying an 8% coupon rate with semiannual payments currently sell at par value. What coupon rate would they have to pay in order to sell at par if they paid their coupons annually? (Hint: what is the effective annual yield on the bond?)3. Two bonds have identical times to maturity and coupon rates. One is callable at 105, the other at 110. Which should have the higher yield to maturity? Why?4. Consider a bond with a 10% coupon and with yield to maturity _ 8%. If the bond’s yield to maturity remains constant, then in 1 year, will the bond price be higher, lower, or unchanged? Why?5. Consider an 8% coupon bond selling for $953.10 with 3 years until maturity making annual coupon payments. The interest rates in the next 3 years will be, with certainty, . Calculate the yield to maturity and realized compound yield of the bond.6. Philip Morris may issue a 10-year maturity fixed-income security, which might include a sinking fund provision and either refunding or call protection.a. Describe a sinking fund provision.b. Explain the impact of a sinking-fund provision on:i. The expected average life of the proposed security.ii. Total principal and interest payments over the life of the proposed security.c. From the investor’s point of view, explain the rationale for demanding a sinking fund provision.7. Bonds of Zello Corporation with a par value of $1,000 sell for $960, mature in 5 years, and have a 7% annual coupon rate paid semiannually.a. Calculate the:i. Current yield.ii. Yield to maturity (to the nearest whole percent, i.e., 3%, 4%, 5%, etc.).iii. Realized compound yield for an investor with a 3-year holding period and a reinvestment rate of 6% over the period. At the end of 3 years the 7% coupon bonds with 2 years remaining will sell to yield 7%.b. Cite one major shortcoming for each of the following fixed-income yield measures:i. Current yield.ii. Yield to maturity.iii. Realized compound yield.9. A 20-year maturity bond with par value of $1,000 makes semiannual coupon payments at a coupon rate of 8%. Find the bond equivalent and effective annual yield to maturity of the bond if the bond price is:a. $950.b. $1,000.c. $1,050.12. Consider a bond paying a coupon rate of 10% per year semiannually when the market interest rate is only 4% per half year. The bond has 3 years until maturity.a. Find the bond’s price today and 6 months from now after the next coupon is paid.b. What is the total (6 month) rate of return on the bond?14. A bond with a coupon rate of 7% makes semiannual coupon payments on January15 and July 15 of each year. The Wall Street Journal reports the asked price for the bond on January 30 at 100:02. What is the invoice price of the bond? The coupon period has 182 days.20. A 30-year maturity, 8% coupon bond paying coupons semiannually is callable in 5 years at a call price of $1,100. The bond currently sells at a yield to maturity of 7% (3.5% per half-year).a. What is the yield to call?b. What is the yield to call if the call price is only $1,050?c. What is the yield to call if the call price is $1,100, but the bond can be called in 2 years instead of 5 years?22. A 2-year bond with par value $1,000 making annual coupon payments of $100 is priced at $1,000. What is the yield to maturity of the bond? What will be the realized compound yield to maturity if the 1-year interest rate next year turns out to be (a) 8%,(b) 10%26. Alarge corporation issued both fixed and floating-rate notes 5 years ago, with terms given in the following table:a. Why is the price range greater for the 9% coupon bond than the floatingrate note?b. What factors could explain why the floating-rate note is not always sold at par value?c. Why is the call price for the floating-rate note not of great importance to investors?d. Is the probability of call for the fixed-rate note high or low?e. If the firm were to issue a fixed-rate note with a 15-year maturity, what coupon rate would it need to offer to issue the bond at par value?f. Why is an entry for yield to maturity for the floating-rate note not appropriate?27. On May 30, 1999, Janice Kerr is considering one of the newly issued 10-year AAA corporate bonds shown in the following exhibit.a. Suppose that market interest rates decline by 100 basis points (i.e., 1%). Contrast the effect of this decline on the price of each bond.b. Should Kerr prefer the Colina over the Sentinal bond when rates are expected to rise or to fall?c. What would be the effect, if any, of an increase in the volatility of interest rates on the prices of each bond?Chapter 152. Which one of the following statements about the term structure of interest rates is true?a. The expectations hypothesis indicates a flat yield curve if anticipated futureshort-term rates exceed current short-term rates.b. The expectations hypothesis contends that the long-term rate is equal to the anticipated short-term rate.c. The liquidity premium theory indicates that, all else being equal, longer maturities will have lower yields.d. The liquidity preference theory contends that lenders prefer to buy securities at the short end of the yield curve.8. Suppose the following table shows yields to maturity of zero coupon U.S. Treasury securities as of January 1, 1996:a. Based on the data in the table, calculate the implied forward 1-year rate of interest at January 1, 1999.b. Describe the conditions under which the calculated forward rate would be an unbiased estimate of the 1-year spot rate of interest at January 1, 1999.c. Assume that 1 year earlier, at January 1, 1995, the prevailing term structure for U.S. Treasury securities was such that the implied forward 1-year rate of interest at January 1, 1999, was significantly higher than the corresponding rate implied by the term structure at January 1, 1996. On the basis of the pure expectations theory of the term structure, briefly discuss two factors that could account for such a decline in the implied forward rate.16. Below is a list of prices for zero-coupon bonds of various maturities.a. An 8.5% coupon $1,000 par bond pays an annual coupon and will mature in 3years. What should the yield to maturity on the bond be?b. If at the end of the first year the yield curve flattens out at 8%, what will be the1-year holding-period return on the coupon bond?21. The yield to maturity (YTM) on 1-year zero-coupon bonds is 5% and the YTM on 2-year zeros is 6%. The yield to maturity on 2-year-maturity coupon bonds with coupon rates of 12% (paid annually) is 5.8%. What arbitrage opportunity is available for an investment banking firm? What is the profit on the activity?22. Suppose that a 1-year zero-coupon bond with face value $100 currently sells at $94.34, while a 2-year zero sells at $84.99. You are considering the purchase of a2-year-maturity bond making annual coupon payments. The face value of the bond is $100, and the coupon rate is 12% per year.a. What is the yield to maturity of the 2-year zero? The 2-year coupon bond?b. What is the forward rate for the second year?c. If the expectations hypothesis is accepted, what are (1) the expected price of the coupon bond at the end of the first year and (2) the expected holding-period return on the coupon bond over the first year?d. Will the expected rate of return be higher or lower if you accept the liquidity preference hypothesis?。
2015 AMC 12A 考题及答案
2015AMC12A考题及答案Problem1What is the value ofProblem2Two of the three sides of a triangle are20and15.Which of the following numbers is not a possible perimeter of the triangle?Problem3Mr.Patrick teaches math to15students.He was grading tests and found that when he graded everyone's test except Payton's,the average grade for the class was80.After he graded Payton's test,the class average became 81.What was Payton's score on the test?Problem4The sum of two positive numbers is5times their difference.What is the ratio of the larger number to the smaller?Problem5Amelia needs to estimate the quantity,where and arelarge positive integers.She rounds each of the integers so that the calculation will be easier to do mentally.In which of these situations will her answer necessarily be greater than the exact value of?Problem6Two years ago Pete was three times as old as his cousin Claire.Two years before that,Pete was four times as old as Claire.In how many years will the ratio of their ages be?Problem7Two right circular cylinders have the same volume.The radius of the second cylinder is more than the radius of the first.What is therelationship between the heights of the two cylinders?Problem8The ratio of the length to the width of a rectangle is:.If therectangle has diagonal of length,then the area may be expressed asfor some constant.What is?Problem9A box contains2red marbles,2green marbles,and2yellow marbles. Carol takes2marbles from the box at random;then Claudia takes2of the remaining marbles at random;and then Cheryl takes the last2marbles. What is the probability that Cheryl gets2marbles of the same color?Problem10Integers and with satisfy.What is?Problem11On a sheet of paper,Isabella draws a circle of radius,a circle of radius ,and all possible lines simultaneously tangent to both circles.Isabella notices that she has drawn exactly lines.How many differentvalues of are possible?Problem12The parabolas and intersect the coordinate axes in exactly four points,and these four points are the vertices of a kite of area.What is?Problem13A league with12teams holds a round-robin tournament,with each team playing every other team exactly once.Games either end with one team victorious or else end in a draw.A team scores2points for every game it wins and1point for every game it draws.Which of the following is NOT a true statement about the list of12scores?Problem14What is the value of for which?Problem15What is the minimum number of digits to the right of the decimal pointneeded to express the fraction as a decimal?Problem16Tetrahedron hasand.What is the volume of the tetrahedron?Problem17Eight people are sitting around a circular table,each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated.What is the probability that no two adjacent people will stand?Problem18The zeros of the function are integers.What is the sum of the possible values of?Problem19For some positive integers,there is a quadrilateral withpositive integer side lengths,perimeter,right angles at and,,and.How many different values of arepossible?Problem20Isosceles triangles and are not congruent but have the same areaand the same perimeter.The sides of have lengths of and,while those of have lengths of and.Which of the followingnumbers is closest to?Problem21A circle of radius passes through both foci of,and exactly four pointson,the ellipse with equation.The set of all possible values of is an interval.What is?Problem22For each positive integer,let be the number of sequences of length consisting solely of the letters and,with no more thanthree s in a row and no more than three s in a row.What is theremainder when is divided by12?Problem23Let be a square of side length1.Two points are chosen independentlyat random on the sides of.The probability that the straight-linedistance between the points is at least is,where andare positive integers and.What is?Problem24Rational numbers and are chosen at random among all rationalnumbers in the interval that can be written as fractions whereand are integers with.What is the probabilitythat is a real number?Problem25A collection of circles in the upper half-plane,all tangent to the-axis,is constructed in layers as yer consists of two circles of radiiand that are externally tangent.For,the circles inare ordered according to their points of tangency with the-axis.For every pair of consecutive circles in this order,a new circle is constructed externally tangent to each of the two circles in the yerconsists of the circles constructed in this way.Let,andfor every circle denote by its radius.What is2015AMC12A Answer Key1.C2.E3.E4.B5.D6.B7.D8.C9.C10.E11.D12.B13.E14.D15.C16.C17.A18.C19.B20.A21.D22.D23.A24.D25.D。
question problem troyble difficuty的 区别
difficulty,trouble,problem,quesfior的区别
difficulty指的是困难，例如你在工作接到了一个大项目，完成它有点困难。
trouble指的是麻烦，苦恼，烦恼。
例如杰克遇到了感情上的烦恼。
problem指的是问题，或者是出了什么问题。
例如汤姆有一个感情的问题。
question指的是一个问题。
我有一个问题问你，problem与question 区别是：problem更广，表示可能出了什么差错。
I have some problems
我有些问题（身体不舒服之类）
I have some questions
我有些疑问（学习方面，不懂的）
The problem is difficult to solve.
The question is difficult to answer.
You are in trouble.
The difficulty is high.
problem (问题->要去解决的)
question (提问->要向别人回答的)
trouble (麻烦) no trouble in doing固定搭配
diffcult(困难) difficulty (难度)
A trouble后面＋in，而difficulty后面跟to。
solve 例句
solve 例句solve 例句如下：1、It isn't going to solve a single thing.这解决不了任何问题。
2、She tried her best to solve the problem.她尽了最大的努力解决这个问题。
3、In itself, it's not a difficult problem to solve.这本身并不是个难解决的问题。
4、I dreamed up a plan to solve both problems at once.我想出了一个可以立刻解决这两个问题的计划。
5、She leaned on him to help her to solve her problems.她依靠他帮忙解决问题。
6、We'll solve the case ourselves and surprise everyone. 我们将自己解决这件事，并让所有人吃惊。
7、They can be entrusted to solve major national problems. 他们能受托解决重大国家问题。
8、The only way to solve homelessness is to provide more homes.解决无家可归的惟一途径是提供更多的收容所。
9、This won't solve the problem but it's a step in the right direction.这虽不能解决问题，却是朝正确方向迈出的一步。
10、Human societies have the power to solve the problems confronting them.人类社会有能力解决他们面临的问题。
11、Their domestic reforms did nothing to solve the problem of unemployment.他们的国内改革没有采取任何措施以解决失业问题。
15的英文怎么拼
15的英文怎么拼15这个词的英文单词是什么?你知道怎么拼写吗?下面是店铺给大家整理的15的英文怎么拼，供大家参阅!15的英文怎么拼fifteen英 [ˌfɪfˈti:n] 美 [fɪfˈtin]15的英语例句1. Fifteen-year-old Danny is on the run from a local authority home.15岁的丹尼从地方当局的收容所逃跑了。
2. They had left school at fifteen and were quite untutored in writing.他们15岁就辍学了，几乎未接受过正式的写作训练。
3. The incident brings the total of people killed to fifteen.这次事故使死亡总人数达到15人。
4. They saw a squadron of fifteen motorcycle policemen driving in V-formation.他们看见15名警察骑着摩托车排成V字形前行。
5. In the basement fifteen employees are busy making bespoke coats.在地下室里，15名员工正在忙着缝制定做的外套。
6. We landed on a grass airstrip, fifteen minutes after leaving Mahe.离开马埃15分钟后，我们降落在一个临时草地机场上。
7. More than fifteen thousand people took part in the memorial service.有15,000多人参加了悼念仪式。
8. Fights with his father lasted anything between fifteen minutes and an hour.和他父亲争吵了大约15分钟到一个小时。
产科护理查房
12月16日 乙肝表面抗体
阳性
03 入院护理诊断/问题
03 护理诊断/问题
12-12 P（problem）：有阴道大量流血的危险 与胎盘位于中央位置有关
12-12 I（intervention）：*垫会阴垫以便观察出血情况，并观察血压、 脉搏、呼吸、面色与阴道流血情况，及早发现 大出血征兆 *进食高蛋白、高维生素、富有含铁的食物， 以纠正贫血。 *多食含粗纤维的食物，保证大便通畅，必要 时遵医嘱给予大便软化剂。
护理诊断及措施 问题与讨论
01 定义
孕28周后若胎盘附着于
，甚
至胎盘下缘达到或覆盖宫颈内口处，其位置低
于胎儿先露部时，称为前置胎盘。前置胎盘是
的常见原因。
01 分 类 边缘性前置胎盘
前置胎盘 完全性前置胎盘
部分性前置胎盘
前置胎盘的病因
受精卵发 育迟缓
子宫内膜发育 不良：多次流 产、刮宫、产 褥感染等
*常规被备葡萄糖酸钙作为解药，严格遵医嘱 严格控制硫酸镁的输入速度及总量。
12-25 O（outcome):住院期间产妇未发生镁离子中毒。
05 知识拓展：
05 定义：
05
凶险性前置胎盘患者往往有剖宫产史以及腹腔脏器手术史， 手术后腹腔粘连、妊娠后胎盘膀胱植入增大再次手术的困难[1]。 凶险性前置胎盘患者出血可发生于产前、产时和产后，且出血迅 速、出血量大，所以凶险性患者临床处理需要针对以上情况，往 往需要多学科的团队合作，包括产科、泌尿外科、新生儿科、麻 醉科、血液科和重症医学科等。根据患者阴道出血量、孕周、生 命体征以及胎儿宫内存活情况等进行个体化处理，现如今临床上 有了新的手术方式---腹主动脉球囊阻断术
3
终止妊娠
期待疗法
施心远主编听力教程4第2Unit12-15问题详解
Unit 12Section I1.急救是一种在专业医疗救护到达之前对意外伤害或事故的受害者所进行的医疗救护。
2.知道如何在紧急情况下进行急救，可能就意味着一个人的生死。
3.一种称为心肺复苏的急救方法，简称CPR，可以挽救心脏病、溺水和休克病人的生命。
4.在美国，CPR训练包括使用防护布或面罩覆盖嘴巴。
5.这有助于防止在做口对口复苏急救时传播疾病。
Section IIDialogueD A B C D C D DPassage1.Life expectancy at birth in the United States in 1901 was 49 years while at the end ofthe century it increased to 77 years.2.Psychologists found that people who entertained positive thoughts regardingthemselves and their future health lived seven years longer than those with negative attitudes.3.We tend to regard medical disorders as the cause of poor health or shortened lifespan,but, really, their significance is small when compared to the impact of the psychological factors on health.4.Exercise, weight loss or non-smoking can increase the lifespan. Another avenue torejuvenation is through creativity.5.The US Census Bureau predicted that the USA would eventually have 5.3 millionpeople aged over 100 in 2100.F T T T T T F TEx.21 It increased from 49 in the year 1901 to 77 at the end of the century, an increase of 57%.2 In human history, preventing early deaths is the main cause of the increase in life expectancy.3 We get older because of three things: aging of arteries, dysfunction of the immune system, and accidents and environmental hazards.4 He/she could potentially feel as young as a 44-year-old.5 The mainstream view is that life expectancy in the US will be in the mid-80s by the year 2050 (up from 77 today) and will top out eventually in the low 90s.Section IIIItem 1:A.a serious snowstorm that hit New York City, US.96 kilometers an hour/localized drifting/68.3 centimeters/blizzard/18691) The road network in many places is impassable. 2) Most of the airports in the region have been closed, with hundreds of flights cancelled. 3) Passengers on transatlantic flightsheading into New York have, in some cases, found themselves diverted to alternative destinations.worked non-stop/keep streets and avenues open/people making their way down Broadway on skisItem 2:A.the evacuation of people living in villages close to V olcano Merapi which is about toerupt.B. T F F T TItem 3:A. the heavy storms and their effects in East Asia.B. landslides/several hundred houses to collapse/the worst flooding/trapped/ missing/landslides and flooding/drowned inside his car/dead in a gutterSection IVPart I:A. Saturday/100/float/tide/senses/effect/aroma/crackling/keep/refuelingB. 1) WaterFire attracts 1 million visitors each year, and brings business to downtownrestaurants, hotels and entertainment venues. It brings more than 33 million dollars in business to the city annually.2) It sets a good example to other riverfront cities.Part IIDictation:1. A new study published in the current issue of American Journal of Public Healthsuggests that men’s behavior may be to blame.2.It has been reported that at every age American males have poorer health and a higherrisk of mortality than females.3.As if that weren’t enough, men tend to work in more dangerous settings than women,and thus account for 90% of on-the-job fatalities, mostly in agriculture.4.In low-lying flood zones, men are more likely to drive around barricades and drown inhigh water.5.These reasons alone would certainly contribute to a shorter life span for men, but theproblem may be even more profound.C A BD D B A DUnit 13Section I6.Avian influenza, or “bird flu”, is a contagious disease of animals caused by viruses thatnormally infect only birds and, less commonly, pigs.7.While all bird species are thought to be susceptible to infection, domestic poultry flocksare especially vulnerable to infections that can rapidly reach epidemic proportions.8.The current outbreak of bird flu is different from earlier ones in that officials have beenunable to contain its spread.9.Rapid elimination of the H5N1 virus among infected birds and other animals isessential to preventing a major outbreak.10.The World Health Organization recommends that infected or exposed flocks ofchickens and other birds be killed in order to help prevent further spread of the virus and reduce opportunities for human infection.1. 禽流感是一种由病毒引起的动物接触性传染病，通常只感染禽类，在少数情况下也会感染猪。
问题的英语短语
问题的英语短语问题，指要求回答或解答的题目。
下面就由店铺为大家带来关于问题的英语短语集锦，希望大家能有所收获。
关于问题的相关短语问题定义 problem definition;问题儿童 problem child;问题教学 problem instruction;问题解答[决] problem solving;问题校验 problem check;问题求解程序 problem solver;问题行为图 problem-behaviour graphs;问题学习 problem learning;问题语言 problem-oriented language关于问题的相关短句1.(需回答的题目) question; problem:solve a problem;解答问题ask a question提问题2.(需研究解决的矛盾等) problem; matter:the heart of the matter; key to the question; key to the solution of a question;问题的关键essence of the problem [question];问题的实质a satchel [a batch] of questions;问题成堆a troublesome problem;难问题questions of common interest;共同关心的问题a key problem;关键问题an ideological problem;思想问题an outstanding issue;悬而未决的问题a question of principle; a matter of principle;原则问题minor issue枝节问题3.(事故或意外) trouble; mishap; sth. wrong:troubleshooter;解决麻烦问题的能手without any mishap;没出什么问题Something is wrong with the machine.那机器有点问题。