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2000-2012美国AMC10中文版试题及答案

2000到2012年AMC10美国数学竞赛0 0P 0 A 0 B 0 C 0D 0 全美中学数学分级能力测验(AMC 10)2000年第01届美国AMC10 (2000年2月日时间75分钟)1. 国际数学奥林匹亚将于2001年在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ⨯M ⨯O =2001，则试问I +M +O 之最大值为。

(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为。

(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是元。

(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直在线移动时，下列各数值有项会变动。

(a) MN 长 (b) △P AB 之周长 (c) △P AB 之面积 (d) ABNM 之面积(A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项 6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在费氏数列各项的个位数字中，最后出现的阿拉伯数字为。

(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分∠ADC ，试问△BDP 之周长为。

amc数学竞赛区区分

amc数学竞赛区区分
AMC数学竞赛主要分为以下几个级别：
1.AMC8：由8年级和8年级以下学生参加。

2.AMC10：由10年级和10年级以下的学生参加。

3.AMC12：由12年级和12年级以下的学生参加。

4.AIME：在AMC10和AMC12测试中成绩达到要求的学生可以参加。

JMO：由AMC10的约前230名美国学生参加。

AMO：由AMC10的约前270名美国学生参加。

其中，AMC8、AMC10、AMC12、AIME、USJMO、USAMO这六个级别比赛组成了AMC系列赛。

AMC数学竞赛的参赛对象和举办时间也不同，具体如下：
1.AMC8：8年级或以下并且挑战当天未满14.5岁的学生可以参加，比赛在每年的11月中举行。

2.AMC10和AMC12：AMC10涵盖9-10年级相关数学内容，包括但不限于基础代数、基础几何、排列组合、初等数论和初等概率等；AMC12
涵盖了整个高中数学课程，包括代数、平面几何、数论证明、排列组合、三角函数、数列和级数、复数和图论等，但不包括微积分。

比赛时间相同，分A、B两次在每年的2月初和2月中举行。

3.AIME：在AMC10和AMC12测试中成绩达到要求的学生可以参加，比赛在每年的3月底举行。

JMO和USAMO：USJMO由AMC10的约前230名美国学生参加，USAMO由AMC10的约前270名美国学生参加，比赛在每年4月的最后一周举行。

总的来说，AMC数学竞赛是一个系列性的数学竞赛，不同级别的比赛难度和参赛对象不同，旨在激发学生对数学的兴趣和才能。

AMC 美国数学竞赛 2000 AMC 10 试题及答案解析

USA AMC 10 20001In the year , the United States will host the International Mathematical Olympiad. Let , , and be distinct positive integers such that the product . What's the largest possible value of the sum ?SolutionThe sum is the highest if two factors are the lowest.So, and .2Solution.3Each day, Jenny ate of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, remained. How many jellybeans were in the jar originally?Solution4Chandra pays an online service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but in January her bill was because she used twice as much connect time as in December. What is the fixxed monthly fee?SolutionLet be the fixed fee, and be the amount she pays for the minutes she used in the first month.We want the fixed fee, which is5Points and are the midpoints of sides and of . As moves along a line that is parallel to side , how many of the four quantities listed below change?(a) the length of the segment(b) the perimeter of(c) the area of(d) the area of trapezoidSolution(a) Clearly does not change, and , so doesn't change either.(b) Obviously, the perimeter changes.(c) The area clearly doesn't change, as both the base and its corresponding height remain the same.(d) The bases and do not change, and neither does the height, so the trapezoid remains the same.Only quantity changes, so the correct answer is .6The Fibonacci Sequence starts with two 1s and each term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in thet units position of a number in the Fibonacci Sequence?SolutionThe pattern of the units digits areIn order of appearance:.is the last.7In rectangle , , is on , and and trisect . What is the perimeter of ?Solution.Since is trisected, .Thus,.Adding, .8At Olympic High School, of the freshmen and of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?There are five times as many sophomores as freshmen.There are twice as many sophomores as freshmen.There are as many freshmen as sophomores.There are twice as many freshmen as sophomores.There are five times as many freshmen as sophomores.SolutionLet be the number of freshman and be the number of sophomores.There are twice as many freshmen as sophomores.9If , where , thenSolution, so ...10The sides of a triangle with positive area have lengths , , and . The sides of a second triangle with positive area have lengths , , and . What is the smallest positive number that is not a possible value of ?SolutionFrom the triangle inequality, and . The smallest positive number not possible is , which is .11Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?SolutionTwo prime numbers between and are both odd.Thus, we can discard the even choices.Both and are even, so one more than is a multiple of four.is the only possible choice.satisfy this, .12Figures , , , and consist of , , , and nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?SolutionSolution 1We have a recursion:.I.E. we add increasing multiples of each time we go up a figure. So, to go from Figure 0 to 100, we add.Solution 2We can divide up figure to get the sum of the sum of the firstodd numbers and the sum of the first odd numbers. If you do not see this, here is the example for :The sum of the first odd numbers is , so for figure , there are unit squares. We plug in to get , which is choice13There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?SolutionIn each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.In the second column from the right, there are two spaces for pegs. One of them is in the same row as the corner peg, so there is only one remaining choice left for the yellow peg in this column.By similar logic, we can fill in the yellow pegs as shown:After this we can proceed to fill in the whole pegboard, so there is only arrangement of the pegs. The answer is14Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were , , , , and . What was the last score Mrs. Walter entered? SolutionThe sum of the first scores must be even, so we must choose evens or the odds to be the first two scores.Let us look at the numbers in mod .If we choose the two odds, the next number must be a multiple of , of which there is none.Similarly, if we choose or , the next number must be a multiple of , of which there is none.So we choose first.The next number must be 1 in mod 3, of which only remains.The sum of the first three scores is . This is equivalent to in mod .Thus, we need to choose one number that is in mod . is the only one that works.Thus, is the last score entered.15Two non-zero real numbers, and , satisfy . Which of the following is a possible value of ?SolutionSubstituting , we get16The diagram shows lattice points, each one unit from its nearest neighbors. Segment meets segment at . Find the length of segment .SolutionSolution 1Let be the line containing and and let be the line containing and . If we set the bottom left point at , then , , , and .The line is given by the equation . The -intercept is , so . We are given two points on , hence we cancompute the slope, to be , so is the lineSimilarly, is given by . The slope in this case is , so . Plugging in the point gives us , so is the line .At , the intersection point, both of the equations must be true, soWe have the coordinates of and , so we can use the distance formula here:which is answer choiceSolution 2Draw the perpendiculars from and to , respectively. As it turns out, . Let be the point on for which ., and , so by AA similarity,By the Pythagorean Theorem, we have ,, and . Let , so , thenThis is answer choiceAlso, you could extend CD to the end of the box and create two similar triangles. Then use ratios and find that the distance is 5/9 of the diagonal AB. Thus, the answer is B.17Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?SolutionConsider what happens each time he puts a coin in. If he puts in a quarter, he gets five nickels back, so the amount of money he has doesn't change. Similarly, if he puts a nickel in the machine, he gets five pennies back and the money value doesn't change. However, if he puts a penny in, he gets five quarters back, increasing the amount of money he has by cents.This implies that the only possible values, in cents, he can have are the ones one more than a multiple of . Of the choices given, the only one is18Charlyn walks completely around the boundary of a square whose sides are each km long. From any point on her path she can see exactly km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?SolutionThe area she sees looks at follows:The part inside the walk has area . The part outside the walk consists of four rectangles, and four arcs. Each of the rectangles has area . The four arcs together form a circle with radius . Therefore the total area she can see is, which rounded to the nearest integer is .19Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the trangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. The ratio of the area of the other small right triangle to the area of the square is SolutionLet the square have area , then it follows that the altitude of one of the triangles is . The area of the other triangle is .By similar triangles, we haveThis is choice(Note that this approach is enough to get the correct answer in the contest. However, if we wanted a completely correct solution, we should also note that scaling the given triangle times changes each of the areas times, and therefore it does not influence the ratio of any two areas. This is why we can pick the side of the square.)20Let , , and be nonnegative integers such that . What is the maximum value of ? SolutionThe trick is to realize that the sum is similar to the product .If we multiply , we get.We know that , therefore.Therefore the maximum value of is equal to the maximum value of . Now we will find this maximum.Suppose that some two of , , and differ by at least . Then this triple is surely not optimal.Proof: WLOG let . We can then increase the value ofby changing and .Therefore the maximum is achieved in the cases where is a rotation of . The value of in this case is . And thus the maximum of is.21If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?I. All alligators are creepy crawlers.II. Some ferocious creatures are creepy crawlers.III. Some alligators are not creepy crawlers.SolutionWe interpret the problem statement as a query about three abstract concepts denoted as "alligators", "creepy crawlers" and "ferocious creatures". In answering the question, we may NOT refer to reality -- for example to the fact that alligators do exist.To make more clear that we are not using anything outside the problem statement, let's rename the three concepts as , , and .We got the following information:▪If is an , then is an .▪There is some that is a and at the same time an .We CAN NOT conclude that the first statement is true. For example, the situation "Johnny and Freddy are s, but only Johnny is a "meets both conditions, but the first statement is false.We CAN conclude that the second statement is true. We know that there is some that is a and at the same time an . Pick one such and call it Bobby. Additionally, we know that if is an , then is an. Bobby is an , therefore Bobby is an . And this is enough to prove the second statement -- Bobby is an that is also a .We CAN NOT conclude that the third statement is true. For example, consider the situation when , and are equivalent (represent the same set of objects). In such case both conditions are satisfied, but the third statement is false.Therefore the answer is .22One morning each member of Angela's family drank an -ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?SolutionThe exact value "8 ounces" is not important. We will only use the fact that each member of the family drank the same amount.Let be the total number of ounces of milk drank by the family and the total number of ounces of coffee. Thus the whole family drank a total of ounces of fluids.Let be the number of family members. Then each family member drank ounces of fluids.We know that Angela drank ounces of fluids.As Angela is a family member, we have .Multiply both sides by to get .If , we have .If , we have .Therefore the only remaining option is .23When the mean, median, and mode of the list are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ? SolutionAs occurs three times and each of the three other values just once, regardless of what we choose the mode will always be .The sum of all numbers is , therefore the mean is .The six known values, in sorted order, are . From this sequence we conclude: If , the median will be . If , the median will be . Finally, if , the median will be .We will now examine each of these three cases separately.In the case , both the median and the mode are 2, therefore we can not get any non-constant arithmetic progression.In the case we have , because. Therefore our three values inorder are . We want this to be an arithmetic progression. From the first two terms the difference must be . Therefore thethird term must be .Solving we get the only solution for this case: . The case remains. Once again, we have ,therefore the order is . The only solution is when , i. e., .The sum of all solutions is therefore .24Let be a function for which . Find the sum of all values of for which .SolutionIn the definition of , let . We get: . As we have , we must have , in other words .One can now either explicitly compute the roots, or use Vieta's formulas. According to them, the sum of the roots ofis . In our case this is .(Note that for the above approach to be completely correct, we should additionally verify that there actually are two distinct real roots. This is, for example, obvious from the facts that and .)25In year , the day of the year is a Tuesday. In year , the day is also a Tuesday. On what day of the week did the of year occur?SolutionClearly, identifying what of these years may/must/may not be a leap year will be key in solving the problem.Let be the day of year , the day of year and the day of year .If year is not a leap year, the day will bedays after . As , that would be a Monday.Therefore year must be a leap year. (Then is days after .) As there can not be two leap years after each other, is not a leap year. Therefore day is days after . We have . Therefore is weekdays before , i.e., is a.(Note that the situation described by the problem statement indeed occurs in our calendar. For example, for we have=Tuesday, October 26th 2004, =Tuesday, July 19th, 2005 and =Thursday, April 10th 2003.)。

2010AMC10美国数学竞赛B卷

2010 AMC10美国数学竞赛B卷1. What is 10010031001003（）（）?---(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 02. Makarla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?(A) 15 (B) 20 (C) 25 (D) 30 (E) 353. A drawer contains red, green, blue, and white socks with at least 2 of each color, What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?(A) 3 (B) 4 (C) 5 (D) 8 (E) 94. For a real number x, define ♡(x) to be the average of x and x2. What is♡(1)+ ♡(2)+ ♡(3)?(A) 3 (B) 6 (C) 10 (D) 12 (E) 205. A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?(A) 2 (B) 3 (C) 4 (D) 5 (E) 66. A circle is centered at O, AB is a diameter and C is a point on the circle with∠COB=50°. What is the degree measure of ∠CAB?(A) 20 (B) 25 (C) 45 (D) 50 (E) 657. A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?(A) 16 (B) 24 (C) 28 (D) 32 (E) 368. A ticket to a school play cost x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for x are possible?(A) 1 (B) 2 (C) 3 (D) 4 (E) 59. Lucky Larry’s teacher asked him to substitute numbers for a, b, c, d, and e in the expression ((()))---+and evaluate the result. Larry ignored the parent thesea b c d ebut added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for a, b, c, and d were 1, 2, 3, and 4, respectively. What number did Larry substitute for e?(A) -5 (B) -3 (C) 0 (D) 3 (E) 510. Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rainin the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?(A) 18 (B) 21 (C) 24 (D) 27 (E) 3011. A shopper plans to purchase an item that has a listed price greater than $100 and can use any one of the three coupons. Coupon A gives 15% off the listed price, Coupon B gives $30 off the listed price, and Coupon C gives 25% off the amount by which the listed price exceeds $100.Let x and y be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is y-x?(A) 50 (B) 60 (C) 75 (D) 80 (E) 10012. At the beginning of the school year, 50% of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and 50% answered “No”. At the end of the school year, 70% answered “Yes” and 30% answered “No”. Altogether, x% of the students gave a difference between the maximum and the minimum possible values of x?(A) 0 (B) 20 (C) 40 (D) 60 (E) 8013. What is the sum of all the solutions of 2602=--?x x x(A) 32 (B)60 (C)92 (D) 120 (E) 12414. The average of the numbers 1, 2, 3, …, 98, 99, and x is 100x. What is x? (A)49101 (B) 50101 (C) 12 (D) 51101 (E) 509915. On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse ’s total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?(A) 25(B) 27 (C) 29 (D) 31 (E) 3316. A square of side length 1 and a circle of radiusshare the same center. What is the area inside the circle, but outside the square?(A)13π- (B) 29π- (C) 18π (D) 14(E) 29π17. Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea ’s score was the median among all students, and hers was the highest score on here team. Andrea ’s teammates Beth and Carla placed 37th and 64th , respectively. How many schools are in the city?(A) 22(B) 23 (C) 24 (D) 25 (E) 2618. Positive integers a, b, and c are randomly and independently selected withreplacement from the set {1, 2, 3, …, 2010}. What is the probability that abc ab a++ is divisible by 3?(A) 13(B) 2981(C) 3181(D) 1127(E) 132719. A circle with center O has area 156π. Triangle ABC is equilateral, BC is a chordon the circle, OA=, and point O is outside △ABC. What is the side length of △ABC?(A)(B) 64 (C) (D) 12 (E) 1820. Two circles lie outside regular hexagon ABCDEF. The first is tangent to AB, and the second is tangent to DE, Both are tangent to lines BC and FA. What is the ratio of the area of the second circle to that of the first circle?(A) 18 (B) 27 (C) 36 (D) 81 (E) 10821. A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?(A) 1/10 (B) 1/9 (C) 1/7 (D) 1/6 (E) 1/522. Seven distinct pieces of candy are to be distributed among three bags. The red bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?(A) 1930 (B) 1931 (C) 1932 (D) 1933 (E) 193423. The entries in a 3×3 array include all the digits from 1 through 9, arranged so thatthe entries in every row and column are in increasing order. How many such arrays are there?(A) 18(B) 24 (C) 36 (D) 42 (E) 6024. A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?(A) 30(B) 31 (C) 32 (D) 33 (E) 3425. Let a>0, and let P(x) be a polynomial with integer coefficients such that(1)(3)(5)(7),(2)(4)(6)(8).P P P P a and P P P P a ========- What is the smallest possible value of a?(A) 105(B) 315 (C) 945 (D) 7! (E) 8!。

2010AMC10A试题及答案解析

2010 AMC 10A problems and solutions.The test was held on February 8, 2010. The first link contains the full set of test problems. The rest contain each individual problem and its solution.Problem 1Mary’s top book shelf holds five books with the follow ing widths, incentimeters: , , , , and . What is the average book width, incentimeters?SolutionTo find the average, we add up the widths , , , , and , to geta total sum of . Since there are books, the average book width isThe answer is .Problem 2Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?SolutionLet the length of the small square be , intuitively, the length of the big square is . It can be seen that the width of the rectangle is .Thus, the length of the rectangle is times large as thewidth. The answer is .Problem 3Tyrone had marbles and Eric had marbles. Tyrone then gavesome of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?SolutionLet be the number of marbles Tyrone gave to Eric. Then,. Solving for yields and . Theanswer is .Problem 4A book that is to be recorded onto compact discs takes minutes toread aloud. Each disc can hold up to minutes of reading. Assumethat the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?SolutionAssuming that there were fractions of compact discs, it would takeCDs to have equal reading time. However, since the number of discs can only be a whole number, there are at least 8 CDs,in which case it would have minutes on each of the 8 discs.The answer is .Problem 5The area of a circle whose circumference is is . What is thevalue of ?SolutionIf the circumference of a circle is , the radius would be . Sincethe area of a circle is , the area is . The answer is .Problem 6For positive numbers and the operation is defined asWhat is ?Solution. Then, is The answer isProblem 7Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?SolutionCrystal first runs North for one mile. Changing directions, she runs Northeast for another mile. The angle difference between North and Northeast is 45 degrees. She then switches directions to Southeast, meaning a 90 degree angle change. The distance now from travellingNorth for one mile, and her current destination is miles, because it is the hypotenuse of a 45-45-90 triangle with side length one (mile). Therefore, Crystal's distance from her starting position, x, is equal to, which is equal to . The answer isTony works hours a day and is paid $per hour for each full yearof his age. During a six month period Tony worked days and earned$. How old was Tony at the end of the six month period?SolutionTony worked hours a day and is paid dollars per hour for eachfull year of his age. This basically says that he gets a dollar for each year of his age. So if he is years old, he gets dollars a day. Wealso know that he worked days and earned dollars. If he wasyears old at the beginning of his working period, he would have earned dollars. If he was years old at the beginningof his working period, he would have earned dollars.Because he earned dollars, we know that he was for someperiod of time, but not the whole time, because then the money earned would be greater than or equal to . This is why he waswhen he began, but turned sometime in the middle and earneddollars in total. So the answer is .The answer is . We could findout for how long he was and . . Then isand we know that he was for days, and for days. Thus,the answer is .Problem 9A palindrome, such as , is a number that remains the same whenits digits are reversed. The numbers and are three-digit andfour-digit palindromes, respectively. What is the sum of the digits ofSolutionis at most , so is at most . The minimum value ofis . However, the only palindrome between and is ,which means that must be .It follows that is , so the sum of the digits is .Marvin had a birthday on Tuesday, May 27 in the leap year . Inwhat year will his birthday next fall on a Saturday?Solution(E) 2017There are 365 days in a non-leap year. There are 7 days in a week. Since 365 = 52 * 7 + 1 (or 365 is congruent to 1 mod 7), the same date (after February) moves "forward" one day in the subsequent year, if that year is not a leap year.For example: 5/27/08 Tue 5/27/09 WedHowever, a leap year has 366 days, and 366 = 52 * 7 + 2. So the same date (after February) moves "forward" two days in the subsequent year, if that year is a leap year.For example: 5/27/11 Fri 5/27/12 SunYou can keep count forward to find that the first time this date falls on a Saturday is in 2017:5/27/13 Mon 5/27/14 Tue 5/27/15 Wed 5/27/16 Fri 5/27/17 Sat Problem 11The length of the interval of solutions of the inequality is. What is ?SolutionSince we are given the range of the solutions, we must re-write the inequalities so that we have in terms of and .Subtract from all of the quantities:Divide all of the quantities by .Since we have the range of the solutions, we can make them equal to .Multiply both sides by 2.Re-write without using parentheses.Simplify.We need to find for the problem, so the answer isProblem 12Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?SolutionThe water tower holds times more water than Logan's miniature. Therefore, Logan should make his towertimes shorter than the actual tower. This ismeters high, or choice .Problem 13Angelina drove at an average rate of kph and then stoppedminutes for gas. After the stop, she drove at an average rate ofkph. Altogether she drove km in a total trip time of hoursincluding the stop. Which equation could be used to solve for the time in hours that she drove before her stop?SolutionThe answer is （）because she drove at kmh for hours (theamount of time before the stop), and 100 kmh for because shewasn't driving for minutes, or hours. Multiplying by gives thetotal distance, which is kms. Therefore, the answer isProblem 14Triangle has . Let and be on and ,respectively, such that . Let be the intersection ofsegments and , and suppose that is equilateral. What isSolutionLet .Since ,Problem 15In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.Brian: "Mike and I are different species."Chris: "LeRoy is a frog."LeRoy: "Chris is a frog."Mike: "Of the four of us, at least two are toads."How many of these amphibians are frogs?SolutionSolution 1We can begin by first looking at Chris and LeRoy.Suppose Chris and LeRoy are the same species. If Chris is a toad, then what he says is true, so LeRoy is a frog. However, if LeRoy is a frog, then he is lying, but clearly Chris is not a frog, and we have a contradiction. The same applies if Chris is a frog.Clearly, Chris and LeRoy are different species, and so we have at least frog out of the two of them.Now suppose Mike is a toad. Then what he says is true because we already have toads. However, if Brian is a frog, then he is lying, yethis statement is true, a contradiction. If Brian is a toad, then what he says is true, but once again it conflicts with his statement, resulting in contradiction.Therefore, Mike must be a frog. His statement must be false, which means that there is at most toad. Since either Chris or LeRoy isalready a toad, Brain must be a frog. We can also verify that his statement is indeed false.Both Mike and Brian are frogs, and one of either Chris or LeRoy is afrog, so we have frogs total.Solution 2Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog.As Mike is a frog, his statement is false, hence there is at most one toad.As there is at most one toad, at least one of Chris and LeRoy is a frog. But then the other one tells the truth, and therefore is a toad. Hence we must have one toad and three frogs.Problem 16Nondegenerate has integer side lengths, is an anglebisector, , and . What is the smallest possible value ofthe perimeter?SolutionBy the Angle Bisector Theorem, we know that . If we use the lowest possible integer values for AB and BC (the measures of AD and DC, respectively), then ,contradicting the Triangle Inequality. If we use the next lowest values (and ), the Triangle Inequality is satisfied. Therefore,our answer is , or choice .Problem 17A solid cube has side length inches. A -inch by -inch square hole iscut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?SolutionSolution 1Imagine making the cuts one at a time. The first cut removes a box . The second cut removes two boxes, each of dimensions, and the third cut does the same as the second cut, on thelast two faces. Hence the total volume of all cuts is .Therefore the volume of the rest of the cube is.Solution 2We can use Principle of Inclusion-Exclusion to find the final volume of the cube.There are 3 "cuts" through the cube that go from one end to the other. Each of these "cuts" has cubic inches. However, we cannot just sum their volumes, as the central cube is included ineach of these three cuts. To get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice.Hence the total volume of the cuts is.Therefore the volume of the rest of the cube is.Solution 3We can visualize the final figure and see a cubic frame. We can find the volume of the figure by adding up the volumes of the edges and corners.Each edge can be seen as a box, and each corner can beseen as a box..Problem 18Bernardo randomly picks 3 distinct numbers from the setand arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the setand also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?SolutionWe can solve this by breaking the problem down into cases andadding up the probabilities.Case : Bernardo picks . If Bernardo picks a then it is guaranteedthat his number will be larger than Silvia's. The probability that he willpick a is .Case : Bernardo does not pick . Since the chance of Bernardopicking is , the probability of not picking is .If Bernardo does not pick 9, then he can pick any number from to .Since Bernardo is picking from the same set of numbers as Silvia, the probability that Bernardo's number is larger is equal to the probability that Silvia's number is larger.Ignoring the for now, the probability that they will pick the samenumber is the number of ways to pick Bernardo's 3 numbers divided by the number of ways to pick any 3 numbers.We get this probability to beProbability of Bernardo's number being greater isFactoring the fact that Bernardo could've picked a but didn't:Adding up the two cases we getProblem 19Equiangular hexagon has side lengthsand . The area of is of the area of thehexagon. What is the sum of all possible values of ?SolutionSolution 1It is clear that is an equilateral triangle. From the Law ofCosines, we get that . Therefore,the area of is .If we extend , and so that and meet at , andmeet at , and and meet at , we find that hexagonis formed by taking equilateral triangle of side lengthand removing three equilateral triangles, , and ,of side length . The area of is therefore.Based on the initial conditions,Simplifying this gives us . By Vieta's Formulas we knowthat the sum of the possible value of is .Solution 2As above, we find that the area of is .We also find by the sine triangle area formula that, and thusThis simplifies to.Problem 20A fly trapped inside a cubical box with side length meter decides torelieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?SolutionThe distance of an interior diagonal in this cube is and the distanceof a diagonal on one of the square faces is . It would not make sense if the fly traveled an interior diagonal twice in a row, as it would return to the point it just came from, so at most the final sum can onlyhave 4 as the coefficient of . The other 4 paths taken can be across a diagonal on one of the faces, so the maximum distance traveled is.Problem 21The polynomial has three positive integer zeros. What is the smallest possible value of ?SolutionBy Vieta's Formulas, we know that is the sum of the three roots ofthe polynomial . Also, 2010 factors into. But, since there are only three roots to the polynomial,two of the four prime factors must be multiplied so that we are left with three roots. To minimize , and should be multiplied, whichmeans will be and the answer is .Problem 22Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point insidethe circle. How many triangles with all three vertices in the interior of the circle are created?SolutionTo choose a chord, we know that two points must be chosen. This implies that for three chords to create a triangle and not intersect at a single point, six points need to be chosen. Therefore, the answer iswhich is equivalent to 28,Problem 23Each of 2010 boxes in a line contains a single red marble, and for , the box in the position also contains white marbles.Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws ared marble. Let be the probability that Isabella stops after drawing exactly marbles. What is the smallest value of for whichSolutionThe probability of drawing a white marble from box is . Theprobability of drawing a red marble from box is .The probability of drawing a red marble at box is thereforeIt is then easy to see that the lowest integer value of that satisfiesthe inequality is .Problem 24The number obtained from the last two nonzero digits of is equal to. What is ?SolutionWe will use the fact that for any integer ,First, we find that the number of factors of in is equal to. Let . The we want is thereforethe last two digits of , or . Since there is clearly anexcess of factors of 2, we know that , so it remains tofind .If we divide by by taking out all the factors of in , we canwrite as where whereevery multiple of 5 is replaced by the number with all its factors of 5 removed. Specifically, every number in the form is replaced by ,and every number in the form is replaced by .The number can be grouped as follows:Using the identity at the beginning of the solution, we can reducetoUsing the fact that (or simply the fact thatif you have your powers of 2 memorized), we can deducethat . Therefore.Finally, combining with the fact that yields.Problem 25Jim starts with a positive integer and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with , then hissequence contains numbers:Let be the smallest number for which J im’s sequence has numbers.What is the units digit of ?SolutionWe can find the answer by working backwards. We begin withon the bottom row, then the goes to the right of the equal'ssign in the row above. We find the smallest value for whichand , which is .We repeat the same procedure except with for the next rowand for the row after that. However, at the fourth row, wesee that solving yields , in which case it would beincorrect since is not the greatest perfect square less than orequal to . So we make it a and solve . We continue onusing this same method where we increase the perfect square until can be made bigger than it. When we repeat this until we have rows,we get:Hence the solution is the last digit of , which is .。

2019年AMC10B美国数学竞赛(真题加详解)

2019 AMC 10B Problems/Problem 1The following problem is from both the 2019 AMC 10B #1 and 2019 AMC 12B #1, so both problems redirect to this page.ProblemAlicia had two containers. The first was full of water and the second was empty. She poured all the water from the first container into the secondcontainer, at which point the second container was full of water. What is the ratio of the volume of the first container to the volume of the secondcontainer?Solution 1Let the first jar's volume be and the second's be . It is giventhat . We find thatWe already know that this is the ratio of the smaller to the larger volumebecause it is less thanSolution 2We can set up a ratio to solve this problem. If is the volume of the firstcontainer, and is the volume of the second container, then:Cross-multiplying allows us to get . Thus the ratio of the volume of the first container to the second containeris .~IronicNinjaSolution 3An alternate solution is to plug in some maximum volume for the firstcontainer - let's say , so there was a volume of in the first container, and then the second container also has a volume of , so youget . Thus the answer is .2019 AMC 10B Problems/Problem 2The following problem is from both the 2019 AMC 10B #2 and 2019 AMC 12B #2, so both problems redirect to this page.ProblemConsider the statement, "If is not prime, then is prime." Which of the following values of is a counterexample to this statement?SolutionSince a counterexample must be value of which is not prime, must be composite, so we eliminate and . Now we subtract from the remaining answer choices, and we see that the only time is not prime iswhen .2019 AMC 10B Problems/Problem 3 ProblemIn a high school with students, of the seniors play a musical instrument, while of the non-seniors do not play a musical instrument. Inall, of the students do not play a musical instrument. How many non-seniors play a musical instrument?Solution 1of seniors do not play a musical instrument. If we denote as the numberof seniors, thenThus there are non-seniors. Since 70% of the non-seniorsplay a musical instrument, .~IronicNinjaSolution 2Let be the number of seniors, and be the number of non-seniors.ThenMultiplying both sides by gives usAlso, because there are 500 students in total.Solving these system of equations give us , .Since of the non-seniors play a musical instrument, the answer issimply of , which gives us .Solution 3 (using the answer choices)We can clearly deduce that of the non-seniors do play an instrument, but,since the total percentage of instrument players is , the non-senior population is quite low. By intuition, we can therefore see that the answer isaround or . Testing both of these gives us the answer . 2019 AMC 10B Problems/Problem 4 ProblemAll lines with equation such that form an arithmeticprogression pass through a common point. What are the coordinates of that point?Solution 1If all lines satisfy the condition, then we can just plug in values for , ,and that form an arithmetic progression. Let's use , , ,and , , . Then the two lines we get are:Use elimination to deduce and plug this into one of the previous line equations. We get Thus the common point is .~IronicNinjaSolution 2We know that , , and form an arithmetic progression, so if the commondifference is , we can say Now wehave , and expandinggives Factoringgives . Since this must always be true (regardless of the values of and ), we musthave and , so and the common point is .2019 AMC 10B Problems/Problem 5 ProblemTriangle lies in the first quadrant. Points , , and are reflected across the line to points , , and , respectively. Assume that none of the vertices of the triangle lie on the line . Which of the following statements is not always true?Triangle lies in the first quadrant.Triangles and have the same area.The slope of line is .The slopes of lines and are the same.Lines and are perpendicular to each other.SolutionLet's analyze all of the options separately.: Clearly is true, because a point in the first quadrant will have non-negative - and -coordinates, and so its reflection, with the coordinates swapped, will also have non-negative - and -coordinates.: The triangles have the same area,since and are the same triangle (congruent). More formally, we can say that area is invariant under reflection.: If point has coordinates , then will have coordinates .The gradient is thus , so this is true. (We know since the question states that none of the points , , or lies on the line , so there is no risk of division by zero).: Repeating the argument for , we see that both lines have slope , so this is also true.: By process of elimination, this must now be the answer. Indeed, ifpoint has coordinates and point has coordinates ,then and will, respectively, have coordinates and . The product of the gradientsof and is , so in fact these lines are never perpendicular to each other (using the "negative reciprocal" condition for perpendicularity).Thus the answer is .CounterexamplesIf and , then the slopeof , , is , while the slope of , ,is . is the reciprocal of , but it is not the negative reciprocal of . To generalize, let denote thecoordinates of point , let denote the coordinates of point ,let denote the slope of segment , and let denote the slope of segment . Then, the coordinates of are , andof are . Then, ,and .If and , , and in these cases, the condition is false.2019 AMC 10B Problems/Problem 6The following problem is from both the 2019 AMC 10B #6 and 2019 AMC 12B #4, so both problems redirect to this page.ProblemThere is a real such that .What is the sum of the digits of ?Solution 1Solving by the quadraticformula,(since clearly ). The answer is therefore .~IronicNinjaSolution 2Dividing both sidesby givesSince is non-negative, . The answer is .Solution 3Dividing both sides by as beforegives . Now factorout , giving . By considering the prime factorization of , a bit of experimentation givesus and , so , so the answeris .2019 AMC 10B Problems/Problem 7The following problem is from both the 2019 AMC 10B #7 and 2019 AMC 12B #5, so both problems redirect to this page.ProblemEach piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either pieces of red candy, pieces of green candy, pieces of blue candy, or pieces of purple candy. A piece of purple candy costs cents. What is the smallest possible value of ?Solution 1If he has enough money to buy pieces of red candy, pieces of green candy, and pieces of blue candy, then the smallest amount of money hecould have is cents. Since a piece of purple candy costs cents, the smallest possible valueof is .~IronicNinjaSolution 2We simply need to find a value of that is divisible by , , and .Observe that is divisible by and , but not . is divisible by , , and , meaning that we have exact change (in this case, cents) to buy each type of candy, so the minimum valueof is .2019 AMC 10B Problems/Problem 8ProblemThe figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?Solution 1We notice that the square can be split into congruent smaller squares, with the altitude of the equilateral triangle being the side of this smaller square. Therefore, the area of each shaded part that resides within a square is the total area of the square subtracted from each triangle (which has already been split in half). When we split an equilateral triangle in half, we gettwo triangles. Therefore, the altitude, which is also the side length of one of the smaller squares, is . We can then compute the areaof the two triangles as .The area of the each small squares is the square of the side length,i.e. . Therefore, the area of the shaded region in each of the four squares is .Since there are of these squares, we multiply this by toget as our answer.Solution 2We can see that the side length of the square is by considering thealtitude of the equilateral triangle as in Solution 1. Using the Pythagorean Theorem, the diagonal of the square isthus . Because of this, the height of one ofthe four shaded kites is . Now, we just need to find the length of that kite. By the Pythagorean Theorem again, this lengthis . Nowusing , the area of one of the four kitesis . 2019 AMC 10B Problems/Problem 9 ProblemThe function is defined by for all real numbers , where denotes the greatest integer less than or equal to the real number . What is the range of ?Solution 1There are four cases we need to consider here.Case 1: is a positive integer. Without loss of generality, assume . Then .Case 2: is a positive fraction. Without loss of generality, assume .Then .Case 3: is a negative integer. Without loss of generality, assume . Then .Case 4: is a negative fraction. Without loss of generality, assume . Then .Thus the range of the function is .~IronicNinja, edited by someone elseSolution 2It is easily verified that when is an integer, is zero. We therefore need only to consider the case when is not an integer.When is positive, , soWhen is negative, let be composed of integer part and fractional part (both ):Thus, the range of f is .Note: One could solve the case of as a negative non-integer in thisway:2019 AMC 10B Problems/Problem 10The following problem is from both the 2019 AMC 10B #10 and 2019 AMC 12B #6, so both problems redirect to this page.ProblemIn a given plane, points and are units apart. How manypoints are there in the plane such that the perimeterof is units and the area of is square units?Solution 1Notice that whatever point we pick for , will be the base of thetriangle. Without loss of generality, letpoints and be and , since for any other combination of points, we can just rotate the plane to makethem and under a new coordinate system. When we pickpoint , we have to make sure that its -coordinate is , because that's the only way the area of the triangle can be .Now when the perimeter is minimized, by symmetry, we put in the middle, at . We can easily see that and will bothbe . The perimeter of this minimal triangleis , which is larger than . Since the minimum perimeter is greater than , there is no triangle that satisfies the condition, givingus .~IronicNinjaSolution 2Without loss of generality, let be a horizontal segment of length .Now realize that has to lie on one of the lines parallel to andvertically units away from it. But is already 50, andthis doesn't form a triangle. Otherwise, without loss ofgenerality, . Dropping altitude , we have a righttriangle with hypotenuse and leg , which is clearly impossible, again giving the answer as .2019 AMC 10B Problems/Problem 11 ProblemTwo jars each contain the same number of marbles, and every marble is either blue or green. In Jar the ratio of blue to green marbles is , and the ratio of blue to green marbles in Jar is . There are green marbles in all. How many more blue marbles are in Jar than in Jar ?SolutionCall the number of marbles in each jar (because the problem specifies that they each contain the same number). Thus, is the number of green marbles in Jar , and is the number of green marbles in Jar .Since , we have , so thereare marbles in each jar.Because is the number of blue marbles in Jar , and is the number of blue marbles in Jar , there are more marbles in Jar than Jar . This means the answer is .2019 AMC 10B Problems/Problem 12 ProblemWhat is the greatest possible sum of the digits in the base-seven representation of a positive integer less than ?Solution 1Observe that . To maximize the sum of the digits, we want as many s as possible (since is the highest value in base ), and this will occur with either of the numbers or . Thus, the answeris .~IronicNinja, edited by some peopleNote: the number can also be , which will also give the answer of . Solution 2Note that all base numbers with or more digits are in fact greaterthan . Since the first answer that is possible using a digit number is , we start with the smallest base number that whose digits sum to ,namely . But this is greater than , so we continue bytrying , which is less than 2019. So the answer is .2019 AMC 10B Problems/Problem 13The following problem is from both the 2019 AMC 10B #13 and 2019 AMC 12B #7, so both problems redirect to this page.ProblemWhat is the sum of all real numbers for which the median of thenumbers and is equal to the mean of those five numbers?SolutionThe mean is .There are three possibilities for the median: it is either , , or .Let's start with .has solution , and the sequenceis , which does have median , so this is a valid solution.Now let the median be .gives , so the sequence is , which has median , so this is not valid.Finally we let the median be ., and the sequence is , which has median . This case is therefore again not valid.Hence the only possible value of is2019 AMC 10B Problems/Problem 14 ProblemThe base-ten representationfor is , where , ,and denote digits that are not given. What is ?Solution 1We can figure out by noticing that will end with zeroes, as there are three s in its prime factorization. Next, we use the fact that is a multiple of both and . Their divisibility rules (see Solution 2) tell usthat and that . By inspection, we see that is a valid solution. Therefore the answer is .Solution 2 (similar to Solution 1)We know that and are both factors of . Furthermore, we knowthat , because ends in three zeroes (see Solution 1). We can simply use the divisibility rules for and for this problem to find and . For to be divisible by , the sum of digits must simply be divisible by .Summing the digits, we get that must be divisible by . Thisleaves either or as our answer choice. Now we test for divisibility by . For a number to be divisible by , the alternating sum must be divisibleby (for example, with the number , ,so is divisible by ). Applying the alternating sum test to this problem, we see that must be divisible by 11. By inspection, we can see that this holds if and . The sumis .2019 AMC 10B Problems/Problem 15ProblemRight triangles and , have areas of 1 and 2, respectively. A side of iscongruent to a side of , and a different side of is congruent to a differentside of . What is the square of the product of the lengths of the other (third)side of and ?Solution 1First of all, let the two sides which are congruent be and , where . The only way that the conditions of the problem can be satisfied is if is the shorter leg of and the longer leg of , and is the longer leg of and thehypotenuse of .Notice that this means the value we are looking for is the squareof , which is just . The area conditions give us twoequations: and .This means that and that .Taking the second equation, we get , sosince , .Since , we get .The value we are looking for is just so the answer is .Solution bySolution 2Like in Solution 1, we have and .Squaring both equations yields and .Let and . Then ,and , so .We are looking for the value of , so the answeris .Solution 3Firstly, let the right triangles be and ,with being the smaller triangle. As in Solution 1,let and . Additionally,let and .We are given that and , sousing , we have and . Dividing the twoequations, we get = , so .Thus is a right triangle, meaningthat . Now by the Pythagorean Theoremin ,.The problem requires the square of the product of the third side lengths of each triangle, which is . By substitution, we seethat = . We alsoknow.Since we want , multiplying both sides by getsus . Now squaringgives .2019 AMC 10B Problems/Problem 16ProblemIn with a right angle at , point lies in the interior of andpoint lies in the interior of so that and the ratio . What is the ratioSolution 1Without loss of generality, let and . Let and .As and areisosceles, and .Then , so isa triangle with .Then , and is a triangle.In isosceles triangles and , drop altitudesfrom and onto ; denote the feet of these altitudesby and respectively. Then by AAA similarity, so we get that ,and . Similarly we get ,and .Solution 2Let , and . (For thissolution, is above , and is to the right of ). Also let ,so , whichimplies . Similarly, , whichimplies . This further impliesthat .Now we seethat. Thus is a right triangle, with side lengths of , , and (by the Pythagorean Theorem, or simply the Pythagorean triple ).Therefore (by definition), ,and . Hence (by thedouble angle formula), giving .By the Law of Cosines in , if , wehaveNow . Thus theanswer is .~IronicNinjaSolution 3Draw a nice big diagram and measure. The answers to this problem are not very close, so it is quite easy to get to the correct answer by simply drawing a diagram. (Note: this strategy should only be used as a last resort!)2019 AMC 12B Problems/Problem 13(Redirected from 2019 AMC 10B Problems/Problem 17)The following problem is from both the 2019 AMC 10B #17 and 2019 AMC 12B #13, so both problems redirect to this page.ProblemA red ball and a green ball are randomly and independently tossed into binsnumbered with the positive integers so that for each ball, the probability that it is tossed into bin is for What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?Solution 1By symmetry, the probability of the red ball landing in a higher-numbered bin is the same as the probability of the green ball landing in a higher-numbered bin. Clearly, the probability of both landing in the same binis (by the geometric series sum formula). Therefore the other two probabilities have to bothbe .Solution 2Suppose the green ball goes in bin , for some . The probability of this occurring is . Given that this occurs, the probability that the red ball goesin a higher-numbered bin is (by thegeometric series sum formula). Thus the probability that the green ball goesin bin , and the red ball goes in a bin greater than , is . Summing from to infinity, we getwhere we again used the geometric series sum formula. (Alternatively, if this sum equals , then by writing out the terms andmultiplying both sides by , we see , which gives .) Solution 3The probability that the two balls will go into adjacent binsisby the geometric series sum formula. Similarly, the probability that the two balls will go into bins that have a distance of from each otheris(again recognizing a geometric series). We can see that each time we add a bin between the two balls, the probability halves. Thus, our answeris , which, by the geometric series sum formula,is .-fidgetboss_4000Solution 4 (quick, conceptual)Define a win as a ball appearing in higher numbered box.Start from the first box.There are possible results in the box: Red, Green, Red and Green, ornone, with an equal probability of for each. If none of the balls is in the first box, the game restarts at the second box with the same kind of probability distribution, so if is the probability that Red wins, we canwrite : there is a probability that "Red" wins immediately,a probability in the cases "Green" or "Red and Green", and in the "None"case (which occurs with probability), we then start again, giving the sameprobability . Hence, solving the equation, we get . Solution 5Write out the infinite geometric series as , . To find the probablilty that red goes in a higher-numbered bin than green, we can simply remove all odd-index terms (i.e term , term , etc.), and then sum theremaining terms - this is in fact precisely equivalent to the method of Solution2. Writing this out as another infinite geometric sequence, we are leftwith . Summing, we get2019 AMC 10B Problems/Problem 18 ProblemHenry decides one morning to do a workout, and he walks of the way from his home to his gym. The gym is kilometers away from Henry's home. At thatpoint, he changes his mind and walks of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks ofthe distance from there back toward the gym. If Henry keeps changing his mindwhen he has walked of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point kilometers from home and a point kilometers fromhome. What is ?Solution 1Let the two points that Henry walks in between be and , with being closer to home. As given in the problem statement, the distances of thepoints and from his home are and respectively. By symmetry, the distance of point from the gym is the same as the distance from home to point . Thus, . In addition, when he walks from point to home, he walks of the distance, ending at point . Therefore, we knowthat . By substituting, we get .Adding these equations now gives . Multiplying by , we get ,so .Solution 2 (not rigorous)We assume that Henry is walking back and forth exactly betweenpoints and , with closer to Henry's home than . Denote Henry's home as a point and the gym as a point .Then and ,so .Therefore,. 2019 AMC 10B Problems/Problem 19The following problem is from both the 2019 AMC 10B #19 and 2019 AMC 12B #14, so both problems redirect to this page.ProblemLet be the set of all positive integer divisors of How many numbers are the product of two distinct elements ofSolutionThe prime factorization of is . Thus, we choose twonumbers and where and, whose product is ,where and .Notice that this is analogous to choosing a divisorof , whichhas divisors. However, some of thedivisors of cannot be written as a product of two distinct divisorsof , namely: , , , and . The last twocannot be so written because the maximum factor of containingonly s or s (and not both) is only or . Since the factors chosen must be distinct, the last two numbers cannot be so written because they would require or . This gives candidatenumbers. It is not too hard to show that every number of the form ,where , and are not both or , can be written asa product of two distinct elements in . Hence the answer is . 2019 AMC 10B Problems/Problem 20The following problem is from both the 2019 AMC 10B #20 and 2019 AMC 12B #15, so both problems redirect to this page.ProblemAs shown in the figure, line segment is trisected bypoints and so that Three semicirclesof radius and have their diameters on and are tangent to line at and respectively. A circle ofradius has its center on The area of the region inside the circle butoutside the three semicircles, shaded in the figure, can be expressed in the form where and are positive integersand and are relatively prime. What is ?SolutionDivide the circle into four parts: the top semicircle (); the bottom sector (), whose arc angle is because the large circle's radius is and the short length (the radius of the smaller semicircles) is , givinga triangle; the triangle formed by the radii of and the chord (), and the four parts which are the corners of a circle inscribedin a square (). Then the area is (in , wefind the area of the shaded region above the semicircles but below the diameter, and in we find the area of the bottom shaded region).The area of is .The area of is .For the area of , the radius of , and the distance of (the smaller semicircles' radius) to , creates two triangles,so 's area is .The area of is .Hence, finding , the desired areais , so the answeris2019 AMC 10B Problems/Problem 21 ProblemDebra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?SolutionWe firstly want to find out which sequences of coin flips satisfy the given condition. For Debra to see the second tail before the seecond head, her first flip can't be heads, as that would mean she would either end with double tails before seeing the second head, or would see two heads before she sees two tails. Therefore, her first flip must be tails. The shortest sequence of flips by which she can get two heads in a row and see the second tail before she sees the secondhead is , which has a probability of . Furthermore, she can prolong her coin flipping by adding an extra , which itself has aprobability of . Since she can do this indefinitely, this gives an infinite geometric series, which means the answer (by the geometric series sum formula)is .Solution 2 (Easier)Note that the sequence must start in THT, which happens with probability. Now, let be the probability that Debra will get two heads in a row after flippingTHT. Either Debra flips two heads in a row immediately (probability ), or flips a head and then a tail and reverts back to the "original position" (probability ). Therefore, , so , so our final answeris . -Stormersyle get rect2019 AMC 10B Problems/Problem 22The following problem is from both the 2019 AMC 10B #22 and 2019 AMC 12B #19, so both problems redirect to this page.ProblemRaashan, Sylvia, and Ted play the following game. Each starts with . A bell rings every seconds, at which time each of the players who currently have money simultaneously chooses one of the other two playersindependently and at random and gives to that player. What is theprobability that after the bell has rung times, each player willhave ? (For example, Raashan and Ted may each decide to give to Sylvia, and Sylvia may decide to give her her dollar to Ted, at which pointRaashan will have , Sylvia will have , and Ted will have , and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their to, and the holdings will be the same at the end of the second round.)SolutionOn the first turn, each player starts off with . Each turn after that, there are only two possibilities: either everyone stays at , which we will writeas , or the distribution of moneybecomes in some order, which we writeas . We will consider these two states separately.In the state, each person has two choices for whom to give their dollar to, meaning there are possible ways that the money canbe rearranged. Note that there are only two ways that we canreach again:1. Raashan gives his money to Sylvia, who gives her money to Ted, who gives his money to Raashan.2. Raashan gives his money to Ted, who gives his money to Sylvia, who gives her money to Raashan.Thus, the probability of staying in the state is , while theprobability of going to the state is (we can check that the 6 other possibilities lead to )In the state, we will label the person with as person A, the person with as person B, and the person with as person C.Person A has two options for whom to give money to, and person B has 2 options for whom to give money to, meaning there are。

澳大利亚数学竞赛AMC-2010C试卷

SATURDAY 7 AUGUST 2010初级卷（7—8 年级）考试时间：75 分钟注意事项一般规定1.未获监考老师许可之前不可翻开此测验题本。

2.各种通讯器材一律不得携入考场，不准使用电子计算器、计算尺、对数表、数学公式等计算器具。

作答时可使用直尺与圆规，以及两面全空白的草稿纸。

3.题目所提供之图形只是示意图，不一定精准。

4.最前25 题为选择题，每题有五个选项。

最后5 题要求填入的答案为0 至999 的正整数。

题目一般而言是依照越来越难的顺序安排，对于错误的答案不会倒扣分数。

5.本活动是数学竞赛而不同于学校测验，别期望每道题目都会作。

考生只与同地区同年级的其它考生评比，因此不同年级的考生作答相同的试卷将不作评比。

6.请依照监考老师指示，谨慎地在答案卡上填写您的基本数据。

若因填写错误或不详所造成之后果由学生自行负责。

7.进入试场后，须等待监考老师宣布开始作答后，才可以打开题本进行答题。

作答须知1.限用B 或2B 铅笔填写答案。

2.请用B 或2B 铅笔在答案卡上将您认为正确选项的圆圈涂满（不是在题本上）。

3.您的答案卡将由计算机阅卷，为避免计算机误判，请不要在答案卡上其它任何地方涂划任何记号。

填写答案卡时，若需要修改，可使用软性橡皮小心擦拭，并确定答案卡上无残留痕迹。

特别约定为确保竞赛之公平性及认证成绩优异学生，AMC 主办单位保留要求考生重测之权利。

─────────────────────────────────────────────────初级卷(7-8 年级)─────────────────────────────────────────────────1-10 题，每题 3 分1. 算式 27＋48－37 等于（A ）32 （B ）38 （C ）48 （D ）52 （E ）68 ─────────────────────────────────────────────────2. 算式22 + 33等于（A ）31 （B ）10 （C ）11 （D ）25 （E ）17 ─────────────────────────────────────────────────3. 右图中，x 之值等于（A ）15 （B ）40 （C ）55 （D ）75 （E ）80─────────────────────────────────────────────────4. 有一堂 55 分钟的课在上午 10：05 结束，请问什么时刻开始上课？（A ）上午 9：15 （B ）上午 9：20 （C ）上午 9：10（D ）上午 9：50 （E ）上午 10：50─────────────────────────────────────────────────5. 算式 2010－20.10 等于（A ）1990.09 （B ）1990.9 （C ）1989.09 （D ）1989.9 （E ）1998.9 ───────────────────────────────────────────────── 6. 算式4 + 1 - 2 等于6 3 （A ） 3 5 6 （B ） 3 2 3 （C ） 4 1 3 （D ） 38 9 （E ） 3 1 2 ───────────────────────────────────────────────── 7. 下图中灰色的两块磁砖占大矩形的1 5 1 。

澳洲AMC数学竞赛2023澳洲AMC真题 D级

犀牛国际教育独家真题澳大利亚AMC 真题D 级Intermediate DivisionQuestions 1 to 10 are worth 3 marks each.1-10题，每题3分1.A 40-minute lesson started at 10:50 am. Exactly half-way through the lesson the fire alarm went off.At what time did the fire alarm go off?一节时长为40分钟的课程从上午10:50开始上课。

当课程正好进行到一半时，火警报警器响了，那么火警报警器响起的时间是？（A ）10:30am （B ）11:00am （C ）11:10am （D ）11:20am （E ）11:30am2.Two rectangles have a vertex in common, as shown. What is the size of the angle marked x °between them?如下方图片，两个长方形共用一个顶点。

那么这两个长方形之间标记为x °角的度数是多少？（A ）10°（B ）20°（C ）30°（D ）40°（E ）50°3.What is the value of ++++789234? ++++789234的值是多少？（A ）61（B ）72（C ）83（D ）94（E ）214.How many 25cm×25cm squares fit in a 50cm×1m rectangle?有多少个边长为25cm 的正方形可以完全放入宽50cm 长1m 的长方形？（A ）1（B ）2（C ）4（D ）6（E ）85.Which one of these is equal to 57×953?57×953的结果是？（A ）321（B ）4321（C ）54321（D ）654321（E ）76543216.A parallelogram PQRS has an area of 60 cm 2 and side PQ of length 10 cm.Which length is 6cm?如下图，平行四边形PQRS的面积为60cm2，边PQ的长度为10cm。

2021年AMC10美国数学竞赛A卷附中文翻译和答案

2011AMC10美国数学竞赛A卷欧阳光明（2021.03.07）1. A cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?(A) $24.00(B) $24.50(C) $25.50(D) $28.00(E) $30.002. A small bottle of shampoo can hold 35 milliliters of shampoo, Whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?(A) 11(B) 12(C) 13(D) 14(E) 153. Suppose [a b] denotes the average of a and b, and {a b c} denotes the average of a, b, and c. What is {{1 1 0} [0 1] 0}?(A)(B)(C)(D)(E)4. Let X and Y be the following sums of arithmetic sequences:X= 10 + 12 + 14 + …+ 100.Y= 12 + 14 + 16 + …+ 102.What is the value of ?(A) 92(B) 98(C) 100(D) 102(E) 1125. At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively.There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?(A) 12(B) (C) (D) 13 (E) 146. Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A∪B, the union of A and B?(A) 5(B) 15(C) 20(D) 35 (E) 3007. Which of the following equations does NOT have a solution?(A) (B) (C)(D) (E)8. Last summer 30% of the birds living on TownLake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?(A) 20(B) 30(C) 40(D) 50(E) 609. A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d, where a, b, c, and d are all positive numbers. Which of the following represents the area of this region?(A) ac + ad + bc + bd(B) ac – ad + bc – bd(C) ac + ad – bc – bd(D) –ac –ad + bc + bd(E) ac – ad – bc + bd10. A majority of the 20 students in Ms. Deameanor’s class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each studentbought, and the total cost of all the pencils was $17.71. What was the cost of a pencil in cents?(A) 7(B) 11(C) 17(D) 23 (E) 7711. Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7·EB. What is the ratio of the area of EFGH to the area of ABCD?(A)(B)(C)(D)(E)12. The players on a basketball team made some three-point shots, some two-point shots, some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team’s total score was 61 points. How many free throws did they make?(A) 13(B) 14(C) 15(D) 16(E) 1713. How many even integers are there between 200 and 700 whose digits are all different and come from the set {1, 2, 5, 7, 8, 9}?(A) 12(B)20(C)72(D) 120 (E) 20014. A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the nume rical value of the circle’s circumference?(A)(B)(C)(D)(E)15. Roy bought a new battery-gasoline hybrid car. On a trip the car ranexclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?(A) 140(B) 240(C) 440(D) 640(E) 84016. Which of the following in equal to ?(A)(B)(C)(D)(E)17. In the eight-term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A + H?(A) 17(B) 18(C) 25(D) 26 (E) 4318. Circles A, B, and C each have radius 1. Circles A and B share one point of tangency. Circle C has a point of tangency with the midpoint of AB. What is the area inside Circle C but outside Circle A and Circle B?(A)(B)(C)(D)(E)19. In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town’s population during this twenty-year period?(A) 42(B) 47(C) 52(D) 57 (E) 6220. Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r isdrawn in a clockwise direction. What is the probability that the two chords intersect?(A)(B)(C)(D)(E)21. Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?(A)(B)(C)(D)(E)22. Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?(A) 2500(B) 2880(C) 3120(D) 3250 (E) 375023. Seven students count from 1 to 1000 as follows:·Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is Alice says 1, 3, 4, 6, 7, 9, …, 997, 999, 1000.·Barbara says all of the numbers that Alice doesn’t say, except she also skips the middle number in each consecutive grope of three numbers.·Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive groupof three numbers.·Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.·Finally, George says the only number that no one else says.What number does George say?(A) 37(B) 242(C) 365(D) 728 (E) 99824. Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?(A)(B)(C)(D)(E)25. Let R be a square region and an integer. A point X in the interior of R is called n-ray partitional if there are n rays emanating from X that divide R into N triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?(A) 1500(B) 1560(C) 2320(D) 2480 (E) 25002011AMC10美国数学竞赛A卷1. 某通讯公司手机每个月基本费为20美元, 每传送一则简讯收5美分（一美元=100 美分）。

AMC10美国数学竞赛A卷附中文翻译和答案

AMC10 美国数学竞赛 A 卷附中文翻译和答案简介AMC10 是美国的一项全国性数学竞赛，主要面向高中学生。

此文档提供了 AMC10 美国数学竞赛 A 卷的问题、答案及中文翻译，帮助考生更好地理解和准备竞赛。

问题和答案问题 1让我们从一个整数开始，每一步都按照以下规则进行操作：如果当前的数是偶数，将它除以 2；如果当前的数是奇数，将它乘以 3 并加 1。

通过这样的操作，我们可以生成一个数列，例如，从 9 开始的数列如下所示：9，28，14，7，22，11，34，...。

显然，这个数列最终会包含两个连续的 1。

以下哪个数开始操作后会生成包含两个连续的 1？$\\textbf{(A)}\\ 111 \\qquad \\textbf{(B)}\\ 120 \\qquad \\textbf{(C)}\\ 125 \\qquad \\textbf{(D)}\\ 130 \\qquad\\textbf{(E)}\\ 139$$\\textbf{(D)}\\ 130$问题 2如果 $2^x \\times 5^y = 5000000$，那么x+x的值是多少？$\\textbf{(A)}\\ 6 \\qquad \\textbf{(B)}\\ 7 \\qquad\\textbf{(C)}\\ 10 \\qquad \\textbf{(D)}\\ 12 \\qquad\\textbf{(E)}\\ 15$答案 2$\\textbf{(C)}\\ 10$问题 3数轴上三个数x、x、x的平均值是 6。

给定x−x=8，x−x=12，那么x的值是多少？$\\textbf{(A)}\\ -10 \\qquad \\textbf{(B)}\\ -6 \\qquad \\textbf{(C)}\\ 0 \\qquad \\textbf{(D)}\\ 6 \\qquad\\textbf{(E)}\\ 10$答案 3$\\textbf{(B)}\\ -6$某菜市场的一个南瓜重 100 磅，其中 99% 的重量是水分。

2008 AMC10美国数学竞赛B卷

2008 AMC10美国数学竞赛B卷1. A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?(A) 2 (B) 3 (C) 4 (D) 5 (E) 62. A 4×4 block of calendar dates has the numbers 1 through 4 in the first row, 8 through 11 in the second, 15 through 18 in the third, and 22 through 25 in the fourth. The order of the numbers in the second and the fourth rows are reversed. The numbers on each diagonal are added. What will be the positive difference between the diagonal sums?(A) 2 (B) 4 (C) 6 (D) 8 (E) 103. Assume that x is a positive real number. Which is equivalent to(A) 16x(B) 14xxx(C)34x(D)12x(E) x4. A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players’ salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player?(A) 270,000 (B) 385,000 (C) 400,000 (D) 430,000 (E) 700,0005. For real numbers a and b, define 2$()aa b a b =-. What is 22()$()x y y x --?(A) 0(B) 22x y + (C) 22x (D) 22y (E) xy6. Points B and C lie on AD. The length of AB is 4 times the length of BD, and the length of AC is 9 times the length of CD. The length of BC is what fraction of the length of AD? (A)136 (B) 113 (C) 110 (D) 536 (E) 157. An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side length 1. How many small triangles are required?(A) 10(B) 25 (C) 100 (D) 250 (E) 10008. A class collects $50 to buy flowers for a classmate who is in the hospital. Roses cost $3 each, and carnations cost $2 each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50?(A) 1(B) 7 (C) 9 (D) 16 (E) 179. A quadratic equation 220ax ax b -+= has two real solutions . What is the average of these two solutions?(A) 1(B) 2 (C) b a (D) 2b a (E)10. Points A and B are on a circle of radius 5 and AB=6. Point C is the midpoint of theminor arc AB. What is the length of the line segment AC?(A)(B) 72 (C) (D) (E) 411. Suppose that n μ is a sequence of real numbers satisfying 212n n n μμμ++=+, and that 39μ= and 6128μ=. What is 5μ? (A) 40(B) 53 (C) 68 (D) 88 (E) 10412. Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?(A) 2500(B) 3000 (C) 3500 (D) 4000 (E) 450013. For each positive integer n, the mean of the first n terms of a sequence is n. What is the 2008th term of the sequence?(A) 2008(B) 4015 (C) 4016 (D) 4,030,056 (E) 4,032,06414. Triangle OAB has O=(0,0), B=(5,0), and A in the first quadrant. In addition, ∠ABO=90° and ∠AOB=30°. Suppose that OA is rotated 90°counterclockwise about O. What are the coordinates of the image of A?(A)((B)((C)(D) (E)15. How many right triangles have integer leg lengths a and b and a hypotenuse of length b+1, Where b<100?(A) 6 (B) 7 (C) 8 (D) 9 (E) 1016. Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)(A) 3/8 (B) 1/2 (C) 43/72 (D) 5/8 (E) 2/317. A poll shows that 70% of all voters approve of the mayor’s work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor’s work?(A) 0.063 (B) 0.189 (C) 0.233 (D) 0.333 (E) 0.44118. Bricklayer Brenda would take nine hours to build a chimney alone, and bricklayer Brandon would take 10 hours to build it alone. When they work together, they talk a lot, and their combined output decreases by 10 bricks per hour. Working together, they build the chimney in 5 hours. How many bricks are in the chimney?(A) 500 (B) 900 (C) 950 (D) 1000 (E) 190019. A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth of 2 feet. What is the volume of water, in cubic feet? (A)24π-(B) 24π-(C) 36π-(D)36π-(E) 48π-20. The faces of a cubical die are marked with the numbers 1,2,2,3,3,4. The faces of another die are marked with the numbers 1,3,4,5,6, and 8. What is the probability that the sum of the top two numbers will be 5,7, or 9?(A) 518(B) 718(C) 1118(D) 34(E) 8921. Ten chairs are evenly spaced around a round table. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?(A) 240 (B) 360 (C) 480 (D) 540 (E) 72022. Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?(A) 112(B) 110(C) 16(D) 13(E) 1223. A rectangular floor measures a by b feet, where a and b are positive integers withb>a. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 footaround the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair (a, b)?(A)1 (B) 2 (C) 3 (D) 4 (E) 524. Quadrilateral ABCD has AB=BC=CD, angle ABC=70 and angle BCD=170. What is the measure of angle BAD?(A) 75 (B) 80 (C) 85 (D) 90 (E) 9525. Michael walks at the rate of 5 feet per second on a long straight path. Trash pails are located every 200feet along the path. A garbage truck travels at 10 feet per second in the same direction as Michael and stops for 20 seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?(A) 4 (B) 5 (C) 6 (D) 7 (E) 8。

amc10考试范围

amc10考试范围
ACM-10 考试是 ACM-10 竞赛的一部分,竞赛由国际计算机程序设计竞赛委员会(ACM)颁发。

ACM-10 竞赛的内容主要包括算法和数据结构,计算机网络,操作系统,数据库,编译原理,计算机图形学,数字图像处理,软件工程等方面。

ACM-10 考试涵盖的内容包括但不限于以下内容:
1. 算法与数据结构:选择、排序、查找、链表、栈、队列、树、图等。

2. 计算机网络:协议、网络拓扑、网络安全、网络设备、路由和交换等。

3. 操作系统:进程管理、内存管理、文件系统、系统调用、线程等。

4. 数据库:关系型数据库、非关系型数据库、SQL 等。

5. 编译原理:编译器、代码生成、静态分析、动态分析等。

6. 计算机图形学:二维和三维图像处理、计算机视觉、图形学算法等。

7. 数字图像处理:图像处理、图像增强、图像分割、图像识别等。

8. 软件工程:软件开发方法、软件测试、软件设计、软件质量保证等。

ACM-10 考试的具体内容会根据每年的竞赛要求和考试机构的不同而有所变化,建议考生通过 ACM-10 竞赛官方网站或相关的考试培
训资料了解最新的考试内容和考试形式。

2021AMC考试真题

2021AMC10A/12ACompiled by Dan Li目录12021AMC10A2 22021AMC12A1312021AMC10A2021AMC10A1.What is the value of(22−2)−(32−3)+(42−4)?A.1B.2C.5D.8E.122.Portia’s high school has3times as many students as Lara’s high school.The two high schoolshave a total of2600students.How many students does Portia’s high school have?A.600B.650C.1950D.2000E.20503.The sum of two natural numbers is17,402.One of the two numbers is divisible by10.If theunits digit of that number is erased,the other number is obtained.What is the difference of these two numbers?A.10,272B.11,700C.13,362D.14,238E.15,4264.A cart rolls down a hill,traveling5inches the first second and accelerating so that duringeach successive1-second time interval,it travels7inches more than during the previous 1-second interval.The cart takes30seconds to reach the bottom of the hill.How far,in inches,does it travel?A.215B.360C.2992D.3195E.32425.The quiz scores of a class with k>12students have a mean of8.The mean of a collection of12of these quiz scores is14.What is the mean of the remaining quiz scores in terms of k?A.14−8 k−12B.8k−168 k−12C.1412−8kD.14(k−12)k2E.14(k−12)8k6.Chantal and Jean start hiking from a trailhead toward a fire tower.Jean is wearing a heavybackpack and walks slower.Chantal starts walking at4miles per hour.Halfway to the tower,the trail becomes really steep,and Chantal slows down to2miles per hour.After reaching the tower,she immediately turns around and descends the steep part of the trail at 3miles per hour.She meets Jean at the halfway point.What was Jean’s average speed,in miles per hour,until they meet?A.12 13B.1C.13 12D.24 13E.27.Tom has a collection of13snakes,4of which are purple and5of which are happy.Heobserves thatI.all of his happy snakes can addII.none of his purple snakes can subtract,andIII.all of his snakes that can’t subtract also can’t add.Which of these conclusions can be drawn about Tom’s snakes?A.Purple snakes can add.B.Purple snakes are happy.C.Snakes that can add are purple.D.Happy snakes are not purple.E.Happy snakes can’t subtract.8.When a student multiplied the number66by the repeating decimal,1.ababab···=1.abWhere a and b are digits,he did not notice the notation and just multiplied66times1.ab.Later he found that his answer is0.5less than the correct answer.What is the2-digit integer ab?A.15B.30C.45D.60E.759.What is the least possible value of(xy−1)2+(x+y)2 for real numbers x and y?A.0B.1 4C.1 2D.1E.210.Which of the following is equivalent to(2+3)(22+32)(24+34)(28+38)(216+316)(232+332)(264+364)A.3127+2127B.3127+2127+2·363+3·263C.3128−2128D.3128+2128E.512711.For which of the following integers b is the base-b number2021b−221b not divisible by3?A.3B.4C.6D.7E.812.Two right circular cones with vertices facing down as shown in the figure below contain thesame amount of liquid.The radii of the top of the liquid surfaces are 3cm and 6cm.Into each cone is dropped a spherical marble of radius 1cm,which sinks to the bottom and is completely submerged without spilling any liquid.What is the ratio of the rise of the liquid level in the narrow cone to the liquid level in the wide cone?A.1:1B.47:43C.2:1D.40:13E.4:113.What is the volume of tetrahedron ABCD with edge lengths AB =2,AC =3,AD =4,BC =√13,BD =2√5,and CD =5?A.1B.2√3C.4D.3√3E.614.All the roots of polynomialz6−10z5+Az4+Bz3+Cz2+Dz+16are positive integers,possibly repeated.What is the value of B?A.−88B.−80C.−64D.−41E.−4015.Values for A,B,C,and D are to be selected from{1,2,3,4,5,6}without replacement(i.e.,no two letters have the same value).How many ways are there to make such choices so that the two curves y=Ax2+B and y=Cx2+D intersect?(The order in which the curves are listed does not matter;for example,the choices A=3,B=2,C=4,D=1is considered the same as the choices A=4,B=1,C=3.D=2.)A.30B.60C.90D.180E.36016.In the following list of numbers,the integer n appears n times in the list for1≤n≤2001,2,2,3,3,3,4,4,4,4,...,200,200,...,200What is the median of the numbers in this list?A.100.5B.134C.142D.150.5E.16717.Trapezoid ABCD has AB ∥CD ,BC =CD =43,and AD ⊥BD .Let O be the intersectionof the diagonals AC and BD ,and let P be the midpoint of BD Given that OP =11,the length AD can be written in the form m √n ,where m and n are positive integers,and n is not divisible by the square of any prime.What is m +n ?A.65B.132C.157D.194E.21518.Let f be function defined on the set of positive rational numbers with the property thatf (a ·b )=f (a )+f (b )for all positive rational numbers a and b .Suppose the f also has the property that f (p )=p for every prime number p .For which of the following numbers x is f (x )<0.A.1732B.1116C.79D.76E.251119.The area of the region bounded by the graph ofx 2+y 2=3|x −y |+3|x +y |is m +nπ,where m and n are integers.What is m +n ?A.18B.27C.36D.45E.5420.In how many ways can the sequence 1,2,3,4,5be rearranged so that no three consecutiveterms are increasing and no three consecutive terms are decreasing?A.10B.18C.24D.32E.4421.Let ABCDEF be an equiangular hexagon.The lines AB ,CD ,and EF determine a triangle with area 192√3,and the lines BC ,DE ,F A determine a triangle with area 324√3.The perimeter of hexagon ABCDEF can be expressed as m +n √p ,where m ,n ,and p are positiveintegers and p is not divisible by the square of any prime.What is m +n +p ?A.47B.52C.55D.58E.6322.Hiram’s algebra notes are 50pages long and are printed on 25sheets of paper:the first sheetcontains pages 1and 2the second sheet contains pages 3and 4.and so on.One day he leaves his notes on the table before leaving for lunch.and his roommate decides to borrow some pages from the middle of the notes.When Hiram comes back,he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean)of the page numbers on all remaining sheets is exactly 19.How many sheets were borrowed?A.10B.13C.15D.17E.2023.Frieda the frog begins a sequence of hops on a3×3grid of squares,moving one square oneach hop and choosing at random the direction of each hop up,down,left,or right.She does not hop diagonally.When the direction of a hop would take Frieda off the grid,she“wraps around”and jumps to the opposite edge.For example If Frieda begins in the center square and makes two hops“up”,the first hop would place her In the top row middle square,and the second hop would cause Frieda to Jump to the opposite edge,landing in the bottom row middle square.Suppose Frieda starts from the center square,makes at most four hops at random,and stops hopping If she lands on a comer square.What Is the probability that she reaches a corner square on one of the four hops?A.9 16B.5 8C.3 4D.25 32E.13 1624.The interior of a quadrilateral is bounded by the graphs of(x+ay)2=4a2and(ax−y)2=a2,where a is a positive real number.What is the area of this region in terms of a,valid for alla>0?A.8a2 (a+1)2B.4a a+1C.8a a+1D.8a2 a2+1E.8a a2+125.How many ways are there to place3Indistinguishable red chips,3indistinguishable bluechips,and3Indistinguishable green chips in the squares of a3×3grid so that no two chips of the same color are directly adjacent to each other,either vertically or horizontally？A.12B.18C.24D.30E.3622021AMC12A2021AMC12A1.What is the value of21+2+3−(21+22+23)?A.0B.50C.52D.54E.572.Under what conditions is√a2+b2=a+btrue,where a and b are real numbers?A.It is never trueB.It is true if and only if ab=0C.It is true if and only if a+b≤0D.It is true if and only if ab=0and a+b≤0E.It is always true3.The sum of two natural numbers is17,402.One of the two numbers is divisible by10.If theunits digit of that number is erased,the other number is obtained.What is the difference of these two numbers?A.10,272B.11,700C.13,362D.14,238E.15,4264.Tom has a collection of13snakes,4of which are purple and5of which are happy.Heobserves thatI.all of his happy snakes can addII.none of his purple snakes can subtract,andIII.all of his snakes that can’t subtract also can’t add.Which of these conclusions can be drawn about Tom’s snakes?A.Purple snakes can add.B.Purple snakes are happy.C.Snakes that can add are purple.D.Happy snakes are not purple.E.Happy snakes can’t subtract.5.When a student multiplied the number66by the repeating decimal,1.ababab···=1.˙abWhere a and b are digits,he did not notice the notation and just multiplied66times1.ab.Later he found that his answer is0.5less than the correct answer.What is the2-digit integer ab?A.15B.30C.45D.60E.756.A deck of cards has only red cards and black cards.The probability of a randomly chosencard being red is1/3.When4black cards are added to the deck,the probability of choosing red becomes1/4.How many cards were in the deck originally?A.6B.9C.12D.15E.187.What is the least possible value of(xy−1)2+(x+y)2for real numbers x and y?A.0B.1 4C.1 2D.1E.28.A sequence of numbers is defined by D0=0,D1=0,D2=1and D n=D n−1+D n−3for n≥3.What are the parities(evenness or oddness)of the triple of numbers(D2021,D2022,D2013), where E denotes even and O denotes oddA.(O,E,O)B.(E,E,O)C.(E,O,E)D.(O,O,E)E.(O,O,O)9.Which of the following is equivalent to(2+3)(22+32)(24+34)(28+38)(216+316)(232+332)(264+364)A.3127+2127B.3127+2127+2·363+3·263C.3128−2128D.3128+2128E.512710.Two right circular cones with vertices facing down as shown in the figure below contain thesame amount of liquid.The radii of the top of the liquid surfaces are3cm and6cm.Into each cone is dropped a spherical marble of radius1cm,which sinks to the bottom and is completely submerged without spilling any liquid.What is the ratio of the rise of the liquid level in the narrow cone to the liquid level in the wide cone?A.1:1B.47:43C.2:1D.40:13E.4:111.A laser is placed at the point(3,5).The laser beam travels in a straight rry wantsthe beam to hit and bounce off the y-axis,then hit and bounce off the x-axis,the hit the point(7,5).What is the total distance the beam will travel along this path?A.2√10B.5√2C.10√2D.15√2E.10√512.All the roots of polynomialz6−10z5+Az4+Bz3+Cz2+Dz+16are positive integers,possibly repeated.What is the value of B?A.−88B.−80C.−64D.−41E.−4013.Of the following complex number z,which one has the property that z5has the greatest realpart?A.−2B.−√3+iC.−√2+√2iD.−1+√3iE.2i14.What is the value of (∑20k=1log5k3k2)·(∑100k=1log9k25k)?A.21B.100log53C.200log35D.2,200E.21,00015.A choir director must select a group of singers from among his6tenors and8basses.Theonly requirements are that the difference between the numbers of tenors and basses must bea multiple of4,and the group must have at least one singer.Let N be the number of groupsthat could be selected.What is the remainder when N is divided by100?A.47B.48C.83D.95E.9616.In the following list of numbers,the integer n appears n times in the list for1≤n≤2001,2,2,3,3,3,4,4,4,4,...,200,200,...,200What is the median of the numbers in this list?A.100.5B.134C.142D.150.5E.16717.Trapezoid ABCD has AB ∥CD ,BC =CD =43,and AD ⊥BD .Let O be the intersectionof the diagonals AC and BD ,and let P be the midpoint of BD Given that OP =11,the length AD can be written in the form m √n ,where m and n are positive integers,and n is not divisible by the square of any prime.What is m +n ?A.65B.132C.157D.194E.21518.Let f be function defined on the set of positive rational numbers with the property thatf (a ·b )=f (a )+f (b )for all positive rational numbers a and b .Suppose the f also has the property that f (p )=p for every prime number p .For which of the following numbers x is f (x )<0.A.1732B.1116C.79D.76E.251119.How many solutions does the equationsin (π2cos (x ))=cos (π2sin x )have in the closed interval [0,π]A.0B.1C.2D.3E.420.Suppose that on a parabola with vertex V and focus F there exists a point A such thatAF =20and AV =21.What is the sum of all possible values of the length F V ?A.13B.403C.413D.14E.43321.The five solutions to the equation (z −1)(z 2+2z +4)(z 2+4z +6)may be written in theform x k +iy k for 1≤k ≤5,where x k and y k are real.Let ϵbe the unique ellipse that passes through the points (x 1,y 1),(x 2,y 2),(x 3,y 3),(x 4,y 4),and (x 5,y 5).The eccentricityof ϵcan be written as √m n ,where m and n are relatively prime positive integers.What is m +n ?(Recall that the eccentricity of an ellipse ϵis the ratio c a ,where 2a is the length of the major axis of ϵ,and 2c is the distance between its two foci.)A.7B.9C.11D.13E.1522.Suppose that the root of the polynomial x 3+ax 2+bx +c are cos2π7,cos 4π7,and cos 6π7,where angles are in radians.What is abc ?A.−349B.−128C.−3√764D.132E.32823.Frieda the frog begins a sequence of hops on a 3×3grid of squares,moving one square oneach hop and choosing at random the direction of each hop up,down,left,or right.She does not hop diagonally.When the direction of a hop would take Frieda off the grid,she “wraps around”and jumps to the opposite edge.For example If Frieda begins in the center square and makes two hops “up ”,the first hop would place her In the top row middle square,and the second hop would cause Frieda to Jump to the opposite edge,landing in the bottom row middle square.Suppose Frieda starts from the center square,makes at most four hops at random,and stops hopping If she lands on a comer square.What Is the probability that she reaches a corner square on one of the four hops?A.916B.58C.34D.2532E.131624.Semicircle Γhas diameter AB of length 14.Circle Ωlies tangent to AB at a point P and intersects Γat Q and R .If QR =3√3,and ∠QP R =60◦,then the area of △P QR is a √b c ,where a and c are relatively prime positive integers,and b is a positive integer not divisibleby the square of any prime.What is a +b +c ?A.110B.114C.118D.122E.12625.Let d(n)denote the number of positive integers that divide n,including1and n.For example,d(1)=1,d(2)=2,and d(12)=6.(This function is known as the divisor function.)Letf(n)=d(n) 3√n.There is a unique positive integer N such that f(N)>f(n)for all positive integers n=N. what is the sum of the digits of N?A.5B.6C.7D.8E.9。

AMC10常用公式汇总2023

美国AMC 10数学竞赛常用公式汇总几何板块1、平行线分线段成比例定理2、射影定理3、角平分线定理4、利用正弦求三角形面积5、斯图尔特定理代数板块6、韦达定理7、算数平均-几何平均不等式8、二项式定理9、合分比定理10、余数定理数论板块11、孙子定理12、费马小定理假如p是质数，若p不能整除a，则a^(p-1) ≡1（mod p），若p能整除a，则a^(p-1) ≡0（mod p）。

若p是质数，且a,p互质，那么a的(p-1)次方除以p的余数恒等于1。

13、威尔逊定理若p为质数，则p可整除(p-1)!+1。

14、欧几里德算法两个数的最大公约数是指两个数的共有约数中的最大一个，例如，(6, 9) 的最大公约数为3; (10, 15) 的最大公约数为5; (252, 105) 的最大公约数为21.欧几里得算法可以高效地求解两个正整数(a, b) 的最大公约数(greatest common divisor, GCD). 该算法基于如下定理：对于正整数(a, b), 其最大公约数等于(b, c) 的最大公约数，其中，c = a % b; 基于上述思想，可以将该算法写成递归的形式：保证a > b.当a % b == 0 成立时，表明 b 为 a 的约数，且b 为(a,b) 二者的最大公约数，所以返回 b.若a % b != 0 成立时，重复步骤2, 寻找(b, a % b) 的最大公约数。

15、立方和公式立方和的公式是：a³+b³=(a+b)(a²-ab+b²)16、欧拉定理计数板块17、互补计数互补计数，顾名思义就是计算所求集合中补集的元素个数。

典型的例子是找出“至少有n 个”的互补情况，也就是“至多有n-1”。

结合题目中出现的"至多"、“至少”这样的关键词，利用互补的思想，可以使一些计数和概率计算变得更简洁有效。

18、容斥原理两个集合的容斥关系公式：AUB=A+B-A∩B（∩为重合的部分）三个集合的容斥关系公式：AUBUC=A+B+C-A∩B-B∩C-C∩A+A∩B∩C。

中文版amc10教材

中文版amc10教材
关于AMC 10的教材，目前市面上有很多中文版的教材供选择。

一般来说，这些教材会覆盖AMC 10考试所涉及的数学知识点和解题
技巧。

这些教材通常包括数学基础知识的讲解，以及大量的例题和
习题供学生练习。

在这些教材中，一般会包括数学基础知识，比如代数、几何、
概率与统计等内容的讲解。

教材通常会从基础知识入手，逐步深入，帮助学生建立起坚实的数学基础。

此外，教材中也会包括大量的
AMC 10考试类型的习题，帮助学生熟悉考试题型，掌握解题技巧。

一些优质的中文版AMC 10教材还会提供一些解题策略和技巧，
帮助学生更好地应对考试。

此外，还会有一些真题以及模拟试卷，
帮助学生进行系统的复习和练习，以便他们在考试中取得更好的成绩。

总的来说，中文版的AMC 10教材会全面覆盖考试所需的数学知
识点和解题技巧，帮助学生系统地复习和准备考试。

选择适合自己
的教材，认真学习和练习，相信可以在AMC 10考试中取得理想的成绩。

amc数学竞赛规则

amc数学竞赛规则摘要：1.AMC 数学竞赛简介2.AMC 数学竞赛的规则3.AMC 数学竞赛的参赛资格和报名方式4.AMC 数学竞赛的考试内容和形式5.AMC 数学竞赛的奖项设置和评分标准6.AMC 数学竞赛的意义和影响正文：【AMC 数学竞赛简介】AMC（American Mathematics Competitions）是美国数学竞赛的简称，是由美国数学及其应用联合会（American Mathematical Society and its subsidiary, the Mathematical Association of America）主办的一项面向全球中学生的数学竞赛。

AMC 数学竞赛旨在激发学生对数学的兴趣，提高学生的数学能力和解决问题的能力，选拔优秀的数学人才。

【AMC 数学竞赛的规则】AMC 数学竞赛的规则分为三个级别：AMC 8、AMC 10 和AMC 12。

其中，AMC 8 适用于8 年级及以下的学生，AMC 10 适用于10 年级及以下的学生，AMC 12 适用于12 年级及以下的学生。

每个级别的竞赛都分为两个阶段：初赛和决赛。

初赛在每年的11 月举行，决赛在次年的2 月举行。

初赛和决赛的考试时间均为75 分钟，满分均为25 分。

【AMC 数学竞赛的参赛资格和报名方式】任何感兴趣的学生都可以报名参加AMC 数学竞赛，无需特殊的参赛资格。

报名方式通常是通过学校或个人在官方网站上进行报名。

报名费用也因地区而异。

【AMC 数学竞赛的考试内容和形式】AMC 数学竞赛的考试内容涵盖了代数、几何、组合、数论等数学领域的知识。

考试形式为个人赛，初赛和决赛均为选择题，决赛还有证明题。

【AMC 数学竞赛的奖项设置和评分标准】AMC 数学竞赛的奖项设置分为个人奖和团队奖。

个人奖包括：满分奖、杰出成就奖、优异奖和参赛奖。

团队奖包括：团队总分第一名、团队平均分第一名和最佳女子团队奖。

评分标准为：正确答案得1 分，错误答案不得分，不答题不得分。

amc10知识点

amc10知识点一、组合数与排列组合1.组合数：是根据N个不同元素中任取m个元素构成的组合数，用C（N，m）表示。

2.组合的计算公式：C（N，m）=N！（m！（N-m）！）3.全排列：N个不同元素按其相应位置依次排列所有可能形式，记为A（N，N）或N！4.可重复排列：N个不同元素或有重复元素排列所有可能形式，用P（m,n）或mn表示。

5.可重复排列的计算公式：P（m,n）mn＝n^m二、分数运算1.有理数：包括正整数、零、负整数及它们的倒数以及它们的组合。

2.无理数：方程的解或者极限的概念定义的数字，比如根号2。

3.带分数：分子分母分述和用一对括号表示的有理数，如18/5。

6.分数化简：去除最大公因数后得到的简化分数，如24/36化简为2/3。

7.分数相加：分母相同的分数直接相加，分母不同的分数需要先化简分母一致后相加。

8.加减乘除：不同分母的分数可以进行相加、减法、乘法和除法，通常的计算方法是先将分数分别化成分数和整数，再将数字运算得出结果，最后再将Manzan化简。

三、比例1.比例：各部分之间存在着两两之间相同或相似的关系，即各部分间的比例相等或相似，称为比例。

2.等比：等比数列是一种等差数列，其每一项与其前项的倍数为等比。

3.等比数列的求法：a＿＿＿＿＿＝a1q^n-1 其中：a为项数，a1为首项数，q为公约数，n为项数。

4.比率：比率就是比较两个东西间的相对大小，是比较系统中变化关系最基本的概念，是两个数量之间的关系，它描述了两个事物相对运动的加速度和减速度。

5.比率的求法：A：B=A／B ，其中A,B就是相比的两个数量。

四、平均数1.算术平均数：是指一组数值的总和除以这组数值的个数所得的平均数。

三角函数：1.三角函数：是一类基本的函数，属于曲线动力学的基础，主要包括正弦函数、余弦函数和正切函数等。

2.正弦函数：是以度数作为自变量，以正弦值作为值的函数，可以表示一个圆周上的极限的数字变化的规律性。

2014AMC10A试题及简答(繁体中文)

9.
某直角三角形的兩股長分別為 2 3 及 6，它們分別為另一股邊上的高。試問此直角三角形第三邊上的高為多少？
(B) 2 (C) 3 (D) 4 (E) 5 (A) 1 10. 由 a 開始連續五個正整數，其平均數為 b。試問由 b 開始連續五個正整數的平均數為何？ (A) a 3 (B) a 4 (C) a 5 (D) a 6 (E) a 7 11. 有位顧客想買某個器具，他有三張不同的折價券，但只可以用其中一張：折價券 1：如果訂價至少 50 美元，可以少付訂價的 10%；折價券 2：如果訂價至少 100 美元，可以少付 20 美元；折價券 3：如果訂價超過 100 美元，可以少付超過 100 美元部分的 18%。下列哪一個訂價，使用折價券 1 比使用折價券 2、折價券 3 能省更多的錢？ (A) 179.95 美元 (B) 199.95 美元 (C) 219.95 美元 (D) 239.95 美元 (E) 259.95 美元 12. 有一個邊長為 6 的正六邊形，以各頂點為圓心，3 為半徑畫全等的六個弧。如圖所示，在這正六邊形內且在六個扇形外部陰影區域之面積為多少？ (A) 27 3 9 (D) 54 3 12 (B) 27 3 6 (E) 108 3 9 (C) 54 3 18
B C B C B
4. 9. 14. 19. 24.
A B C E C
5. 10. 15. 20. 25.
C B B D B
虛線是從一條長邊的第一個三等分點連到對邊的第二個三等分點。若將紙片沿著此虛線折疊壓平後得到另外一個新的圖形，其總面積為 B，則 B : A 為下列何者？
(A) 1 : 2
(B) 3 : 5
(C) 2 : 3

AMC 美国数学竞赛 2001 AMC 10 试题及答案解析

USA AMC 10 20011The median of the listis . What is the mean?Solution2A number is more than the product of its reciprocal and its additive inverse. In which interval does the number lie?Solution3The sum of two numbers is . Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?Solution4What is the maximum number of possible points of intersection of a circle and a triangle?Solution5How many of the twelve pentominoes pictured below have at least one line of symettry?Solution6Let and denote the product and the sum, respectively, of thedigits of the integer . For example, and . Supposeis a two-digit number such that . What is the units digit of ?Solution7When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?Solution8Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today willthey next be together tutoring in the lab?Solution9The state income tax where Kristin lives is levied at the rate of of the first of annual income plus of any amount above . Kristin noticed that the state income tax she paid amounted to of her annual income. What was her annual income?Solution10If , , and are positive with , , and , then isSolution11Consider the dark square in an array of unit squares, part of which is shown. The ﬁrst ring of squ ares around this center square contains unit squares. The second ring contains unit squares. If we continue this process, the number of unit squares in the ring isSolution12Suppose that is the product of three consecutive integers and that is divisible by . Which of the following is not necessarily a divisor of Solution13A telephone number has the form , where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, , , and . Furthermore, , , and are consecutive even digits; , , , and are consecutive odd digits; and . Find .Solution14A charity sells 140 benefit tickets for a total of . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?Solution15A street has parallel curbs feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is feet and each stripe is feet long. Find the distance, in feet, between the stripes.Solution16The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum?Solution17Which of the cones listed below can be formed from a sector of a circle of radius by aligning the two straight sides?A cone with slant height of and radiusA cone with height of and radiusA cone with slant height of and radiusA cone with height of and radiusA cone with slant height of and radiusSolution18The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest toSolution19Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?Solution20A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?Solution21A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter and altitude , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.Solution22In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by , , , , and . Find .Solution23A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?Solution24In trapezoid , and are perpendicular to , with, , and . What is ?Solution25How many positive integers not exceeding are multiples of or but not ?。

2019年 AMC 10A真题含答案

2019AMC10AProblem1What is the value of下列算式的值是多少Problem2What is the hundreds digit of(20!−15!)的百位数字是多少？Problem3Ana and Bonita are born on the same date in different years,years st year Anawas times as old as Bonita.This year Ana's age is the square of Bonita's age.What isAna和Bonita出生在不同年份的同一天,相隔n年。

去年Ana的年龄是Bonita年龄的5倍。

今年Ana的年龄是Bonita年龄的平方。

问n是多少？Problem4A box contains red balls,green balls,yellow balls,blue balls,white balls,and black balls.What is the minimum number of balls that must be drawn from the box withoutreplacement to guarantee that at least balls of a single color will be drawn一个盒子中有28个红球，20个绿球，19个黄球，13个蓝球，11个白球和9个黑球。

为了保证至少取出15个单一颜色的球,在不允许放回重取的情况下，必须从盒子里取出的球的数量最少是多少个？Problem5What is the greatest number of consecutive integers whose sum is最多可以有多少个连续整数，它们的总和是45?Problem6For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?a squarea rectangle that is not a squarea rhombus that is not a squarea parallelogram that is not a rectangle or a rhombusan isosceles trapezoid that is not a parallelogram在以下类型的四边形中有多少种，在四边形所在平面上存在一个与四边形的所有四个顶点等距的点？•正方形•不是正方形的长方形•不是正方形的菱形•不是长方形或菱形的平行四边形•不是平行四边形的等腰梯形Problem7Two lines with slopes and intersect at.What is the area of the triangle enclosed by thesetwo lines and the line两条斜率分别为和2的直线相交于(2,2)。

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Senior DivisionQuestions 1to 10,3marks each1.The value of 2010−20.10is (A)1990.09(B)1990.9(C)1989.09(D)1989.9(E)1998.92.If m =3and n =−35,then mn equals(A)−5(B)5(C)−95(D)−53(E)153.The midpoint of P Q is M (−4,6).The point Q has coordinates (10,12).The point P is (A)(−18,0)(B)(−18,18)(C)(−10,0)(D)(3,9)(E)(3,18)4.The number 63is 87.5%of which number?(A)45(B)70(C)72(D)74(E)755.What percentage of the largest square is covered by the shaded square?(A)6.25%(B)10%(C)12.5%(D)16%(E)25%.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................6.Seven scores 8,10,24,28,23,9and x ,have the property that the mean andmedian are both x .The value of x is (A)15(B)17(C)19(D)21(E)23S 27.The radius of circle P is 23of the radius of circle Q and the radius of circle Q is34of the radius of circle R ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................PQRWhat fraction of the area of the largest circle R is the shaded area?(A)13(B)59(C)14(D)316(E)5168.A coin is tossed ﬁve times.What is the probability that the result will not be ﬁve tails in a row?(A)1516(B)2732(C)45(D)910(E)31329.A rectangle is divided into x rows of y identical squares.Half of them are shaded to form the border with uniform width of 1square as shown.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................The sum of x and y could be (A)17(B)20(C)18(D)19(E)1610.When the numbers x 3,x 2,x ,−x and √x are arranged in order from the largest tothe smallest for any value of x where 0<x <1,the middle number is(A)x 3(B)x 2(C)x (D)−x (E)√xS 3Questions 11to 20,4marks each11.For all values of x ,the expression 73x +72x72x +7xis equal to(A)49(B)72x(C)7(D)7x(E)112.P QS is a triangle with R lying on QS ,with P Q =P S =SR and QRP =QP S ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................QSPRThe size of P SR ,in degrees,is (A)30(B)36(C)45(D)60(E)7013.If 3a +4b 2a −2b =5,then a 2+2b 2abequals(A)1(B)2(C)3(D)4(E)514.The value of (123456785)×(123456782)−(123456783)×(123456784)is(A)−2(B)−1(C)0(D)1(E)none of these15.The length of each side of a triangle like the one below is a di ﬀerent prime number and its perimeter is also a prime number...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................What is the smallest possible perimeter of such a triangle?(A)11(B)17(C)19(D)23(E)29S 416.The 5-digit number a 986b ,where a is the ﬁrst digit and b is the units digit,isdivisible by 72.What is the value of a +b ?(A)9(B)10(C)12(D)13(E)1517.A cap consists of six pieces,all the same size and shape...............................................................................................................................................................................................................................................................................................................................................................•.........................................................................................................................................................................................................................................................................................................................................................If each piece can be either gold or brown,how many di ﬀerent caps can be made?(A)12(B)14(C)16(D)18(E)2018.For all positive integers n ,Snap(n )=2n if n is even and Snap(n )=3n if n is odd.If p is a prime number greater than 2,what is the value of Snap ≥Snap(p −1)−p ¥?(A)p −2(B)2p −2(C)2(p −2)(D)3p −2(E)3(p −2)19.A circle is inscribed in a quadrant of a larger circle.Theratio of the area of the inner circle to that of the quadrant is(A)2:3(B)4:5(C)3:(2+√3)(D)√2:√3(E)4:(3+2√2)..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................20.The operationNmeans a Nb =a +b 2.If a >0and (aNa )Na =aN(aNa ),then a equals(A)1(B)√2(C)√2−1(D)√2+1(E)2S 5Questions 21to 25,5marks each21.The super factorial number 1!×2!×3!×···×12!can be written as a factorialtimes a perfect square,that is,in the form m !×n 2.What is the value of m ?(A)4(B)6(C)8(D)10(E)1222.The rectangular piece of paper pictured has length AB =24cm and widthAD =10cm.It is folded along the diagonal AC and then triangle ACD is folded along the line AE so that AD is aligned with AC ...............................................................................................................................................................................................................................................................................................................................................................✿..............................................................................A B DC10cm 24cm =⇒...............................................................................................................................................................................................................................................................................................................................................................................................................✇................................A B D CE =⇒..........................................................................................................................................................................................................................................................................................................................................................................................ABDCE=⇒...............................................................................................................................................................................................................................................................................................................................................................................................................................A BDCEHow long,in centimetres,is DE ?(A)132(B)10√3(C)203(D)8(E)1223.There are sixteen di ﬀerent ways of writing four-digit strings using 1s and 0s.Threeof these strings are 1010,0100and 1001.These three can be found as substrings of 101001.There is a string of nineteen 1s and 0s which contains all sixteen strings of length 4exactly once.If this string starts with 1111,the last four digits are(A)1110(B)0000(C)0110(D)1010(E)011124.What is the smallest n such that no matter how n points are placed inside or onthe surface of a cube of side length 16units,there are at least two of these points which are closer than 14units to each other?(A)8(B)9(C)11(D)12(E)13S 625.Loki stands at the centre of a forest which has trees with trunks of identical radiiat every integer coordinate point except the origin,where he is standing.From where he is,he cannot see beyond the second tree in any direction.That is,he cannot see any tree with either coordinate of magnitude greater than 2.What is the smallest possible radius of the tree trunks?(A)12(B)13(C)1√10(D)1√13(E)12(√13−3)For questions 26to 30,shade the answer as an integer from 0to 999inthe space provided on the answer sheet.Question 26is 6marks,question 27is 7marks,question 28is 8marks,question 29is 9marks and question 30is 10marks.26.If m +n =11and m 2+n 2=99,what is the value of m 3+n 3?27.A 3-digit number is subtracted from a 4-digit number and the result is a 3-digitnumber.−=The 10digits are all di ﬀerent.What is the smallest possible result?28.In the triangle P QR ,P Q =P R =40cm and S is a point on QR such thatP S =25cm.The extension of P S meets the circle through P QR at T ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................QPRSTWhat is the length,in centimetres,of P T ?29.A polynomial f is given.All we know about it is that all its coe ﬃcients are non-negative integers,f (1)=6and f (7)=3438.What is the value of f (3)?S730.There are many towns on the island of Tetra,all connected by roads.Each townhas three roads leading to three other diﬀerent towns:one red road,one yellow road and one blue road,where no two roads meet other than at towns.If you start from any town and travel along red and yellow roads alternately(RYRY...) you will get back to your starting town after having travelled over six diﬀerent roads.In fact RYRYRY will always get you back to where you started.In the same way,going along yellow and blue roads alternately will always get you back to the starting point after travelling along six diﬀerent roads(YBYBYB).On the other hand,going along red and blue roads alternately will always get you back to the starting point after travelling along four diﬀerent roads(RBRB).How many towns are there on Tetra?。