美国数学竞赛AMCAB习题及答案

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2019AMCAB卷详解及AIME班课

2019AMCAB卷详解及AIME班课2019年AMC10B及12B本⽇完赛, 应官⽅要求，B卷题不得当⽇内公开讨论，翰林学院将同步公开2019AMC10B和12B卷答案及详解。

预览链接获取⽅式公众号后台回复2019AMC10B或2019AMC12B可获取2019AMC B卷答案，解析及结果发布链接，我们同步跟新2019amc 晋级2019 AIME cutoff 分数线。

2019年美国数学竞赛AMC10/12A第⼀轮考试于2019年2⽉7⽇结束！翰林学院整理了2019AMC10A/12A 真题及答案还有翰林导师的独家解析，供⼤家参考传阅。

2019 AMC 10A 真题及答案详解2019 AMC 10A 真题⾸发⽂字版点击此处下载参考答案和解析（仅供参考）2019 AMC 12A 真题及答案详解2019AMC12A 真题⾸发⽂字版点击此处下载参考答案和解析（仅供参考）AIME II卷中国时间 3⽉22⽇开赛错过寒假AIME考前冲刺班的学员不⽤着急应备考学员和家长的强烈要求翰林已再度增开了新冲刺班！时间不等⼈，赶紧上车！AIME春季考前冲刺班春季⾸班暂定2019年2⽉17⽇开课点击此处快速报名备注：3⼩时/次课，中途会加课⼀次，全班学员时间⼀致可以按照学员要求调整课时；早班时段同样适⽤海外学员。

如需1对1辅导可咨询⼩助⼿微信：linstitute1课程适⽤对象2019AMC12分数或预测分数在90*分及以上的学⽣2019AMC10分数或预测分数在110*分及以上的学⽣*翰林学院预估分数，实际分数以稍后MAA官⽅数据为准点击查看往年参赛标准课程⼤纲授课内容如下，具体以⽼师授课情况为准。

代数部分多项式，代数⼿法进阶，数列，⼆次数列进阶不等式进阶Rearrngement Inequality, Cauchy-Schwarz, Jensonʼs Inequlity and more⼏何欧⽒⼏何的处理⼿法，共点和共线，多边形和圆，radical axis, 射影⼏何⼊门三⾓函数进阶，解析⼏何solving geometric problems using vectors, solving geometric problems in complex plane.⽴体⼏何进阶数论类费布拉奇数列和质数分解的⼏个有趣结论模运算进阶Eulerʼs Toteint Function, Fermatʼs Christmas Theorem, Fermatʼs little Theroem快速筛数Quadatic Residue, Lifting the Exponent Lemma排列组合⼆项式系数的运算 Vandermondeʼs identity⼏种常见的排列组合思维⽅法，图论⼊门Probability and Expectation随机过程 Markov ChainRoots of Unity Filter知识综合运⽤数形结合，构造函数，竞赛技巧和总复习课程导师介绍苟岩北京师范⼤学材料物理本科，布⾥斯托⼤学应⽤数学硕博，国内数学物理竞赛背景，⾼考数学148分，曾负责国内⼀线机构AMC项⽬研发和教师培训，担任国内知名⾼中国际部数学⽼师/竞赛教练。

精选美国amc中文版试题及答案

2000到2012年A M C10美国数学竞赛P 0 A 0B 0 C0 D 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) △PAB 之周长 (c) △PAB 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 美国数学竞赛试题+详解英文版

2013 AMC8 Problems1.Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?2.A sign at the fish market says, "50% off, today only: half-pound packages for just $3 perpackage." What is the regular price for a full pound of fish, in dollars?What is the value of?3.4.Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill? 5.Hammie is in thegrade and weighs 106 pounds. His quadruplet sisters are tiny babiesand weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?6.The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?7.Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?8.A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?9.The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?10.What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?11.Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?12.At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?13.When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?14.Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?15.If , , and , what is the product of , , and ?16.A number of students from Fibonacci Middle School are taking part in a community serviceproject. The ratio of -graders to -graders is , and the the ratio of -graders to-graders is . What is the smallest number of students that could be participating in the project?17.The sum of six consecutive positive integers is 2013. What is the largest of these six integers?18.Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?19.Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?20.A rectangle is inscribed in a semicircle with longer side on the diameter. What is thearea of the semicircle?21.Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?22.Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?23.Angle of is a right angle. The sides of are the diameters of semicirclesas shown. The area of the semicircle on equals , and the arc of the semicircle onhas length . What is the radius of the semicircle on ?24.Squares , , and are equal in area. Points and are the midpointsof sides and , respectively. What is the ratio of the area of the shaded pentagonto the sum of the areas of the three squares?25.A ball with diameter 4 inches starts at point A to roll along the track shown. The track iscomprised of 3 semicircular arcs whose radii are inches, inches, andinches, respectively. The ball always remains in contact with the track and does notslip. What is the distance the center of the ball travels over the course from A to B?2013 AMC8 Problems/Solutions1. ProblemDanica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?Solution:In order to have her model cars in perfect, complete rows of 6, Danica must have a number ofcars that is a multiple of 6. The smallest multiple of 6 which is larger than 23 is 24, so she'll need to buy more model car.2.A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars?ProblemSolution: The 50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound of fish is $6. Consequently, if half a pound of fish costs $6, then a whole pound of fish is dollars.What is the value of?3. ProblemNotice that we can pair up every two numbers to make a sum of 1:SolutionTherefore, the answer is .4. ProblemEight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill.What was the total bill?Each of her seven friends paidto cover Judi's portion. Therefore, Judi's portion mustbe. Since Judi was supposed to payof the total bill, the total bill must be.Solution5.Hammie is in thegrade and weighs 106 pounds. His quadruplet sisters are tiny babiesand weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these fivechildren or the median weight, and by how many pounds?ProblemLining up the numbers (5, 5, 6, 8, 106), we see that the median weight is 6 pounds. SolutionThe average weight of the five kids is .Therefore, the average weight is bigger, bypounds, making the answer.6. The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example,. What is the missing number in the top row?ProblemSolutionLet the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.Solution 1: Working BackwardsWe see that, making.It follows that, so.Another way to do this problem is to realize what makes up the bottommost number. Thismethod doesn't work quite as well for this problem, but in a larger tree, it might be faster. (In this case, Solution 1 would be faster since there's only two missing numbers.)Solution 2: Jumping Back to the StartAgain, let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.We can write some equations:Now we can substitute into the first equation using the two others:7. Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass,Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clearthe crossing at a constant speed. Which of the following was the most likely number of cars inthe train?ProblemIf Trey saw, then he saw.Solution 12 minutes and 45 seconds can also be expressed asseconds.Trey's rate of seeing cars,, can be multiplied byon the top andbottom (and preserve the same rate):. It follows that the most likely number of cars is.2 minutes and 45 seconds is equal to.Solution 2Since Trey probably counts around 6 cars every 10 seconds, there are groups of 6cars that Trey most likely counts. Since, the closest answer choice is.8. A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?ProblemFirst, there areways to flip the coins, in order.Solution The ways to get two consecutive heads are HHT and THH. The way to get three consecutive heads is HHH.Therefore, the probability of flipping at least two consecutive heads is .9. The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then onwhich jump will he first be able to jump more than 1 kilometer?ProblemThis is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters, we note that. SolutionHowever, because the first term isand not, the solution to the problem is10. What is the ratio of the least common multiple of 180 and 594 to the greatest common factorof 180 and 594?ProblemTo find either the LCM or the GCF of two numbers, always prime factorize first. Solution 1The prime factorization of . The prime factorization of .Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is ). Multiply all of these to get 5940.For the GCF of 180 and 594, use the least power of all of the numbers that are in bothfactorizations and multiply. = 18. Thus the answer = =.We start off with a similar approach as the original solution. From the prime factorizations, the GCF is 18.Similar SolutionIt is a well known fact that. So we have,.Dividing by 18 yields .Therefore, .11. Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?ProblemWe use that fact that . Let d= distance, r= rate or speed, and t=time. In this case, letrepresent the time.SolutionOn Monday, he was at a rate of . So,.For Wednesday, he walked at a rate of . Therefore,.On Friday, he walked at a rate of. So,. Adding up the hours yields++=.We now find the amount of time Grandfather would have taken if he walked atperday. Set up the equation,.To find the amount of time saved, subtract the two amounts: -=.To convert this to minutes, we multiply by 60.Thus, the solution to this problem is12. At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?ProblemFirst, find the amount of money one will pay for three sandals without the discount. We have.SolutionThen, find the amount of money using the discount: .Finding the percentage yields .To find the percent saved, we have13. ProblemWhen Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?Let the two digits be and. SolutionThe correct score was . Clara misinterpreted it as. The difference between thetwo iswhich factors into. Therefore, since the difference is a multiple of 9,the only answer choice that is a multiple of 9 is.14.Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?ProblemThe probability that both show a green bean is. The probability that both show ared bean is . Therefore the probability isSolution15. If ,, and , what is the product of, , and ?ProblemSolutionTherefore,.Therefore,.To most people, it would not be immediately evident that , so we can multiply 6'suntil we get the desired number:, so.Therefore the answer is16. A number of students from Fibonacci Middle School are taking part in a community serviceproject. The ratio of-graders to-graders is, and the the ratio of-graders to-graders is . What is the smallest number of students that could be participating inthe project?ProblemSolutionWe multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:Solution 1: AlgebraTherefore, the ratio of 8th graders to 7th graders to 6th graders is. Since the ratiois in lowest terms, the smallest number of students participating in the project is.The number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallest possibility). Then there are 6th graders and7th graders. The numbers ofstudents isSolution 2: Fakesolving17. The sum of six consecutive positive integers is 2013. What is the largest of these six integers?ProblemThe mean of these numbers is. Therefore the numbers are, so the answer isSolution 1Let thenumber be . Then our desired number is.Solution 2Our integers are , so we have that.Let the first term be. Our integers are. We have,Solution 318.Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?ProblemThere arecubes on the base of the box. Then, for each of the 4 layers abovethe bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are4 feet left), there arecubes. Hence, the answer is.Solution 1 We can just calculate the volume of the prism that was cut out of the originalbox. Each interior side of the fort will be 2 feet shorter than each side of the outside. Since thefloor is 1 foot, the height will be 4 feet. So the volume of the interior box is.Solution 2The volume of the original box is . Therefore, the number of blockscontained in the fort is19. Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?ProblemIf Hannah did better than Cassie, there would be no way she could know for sure that she didn't get the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, ifHannah did worse than Bridget, there is no way Bridget could have known that she didn't getthe highest in the class. Therefore, Hannah did better than Bridget, so our order isSolution20. Arectangle is inscribed in a semicircle with longer side on the diameter. What is thearea of the semicircle?ProblemSolutionA semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem,. The area is21. ProblemSamantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?SolutionThe number of ways to get from Samantha's house to City Park is, and the number ofways to get from City Park to school is. Since there's one way to go through CityPark (just walking straight through), the number of different ways to go from Samantha's house to City Park to school22.Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?ProblemThere are 61 vertical columns with a length of 32 toothpicks, and there are 33 horizontal rowswith a length of 60 toothpicks. An effective way to verify this is to try a small case, i.e. a grid of toothpicks. Thus, our answer isSolution23.Angleof is a right angle. The sides ofare the diameters of semicircles as shown. The area of the semicircle on equals, and the arc of the semicircle onhas length . What is the radius of the semicircle on?ProblemIf the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagorean theorem says that the other side has length 15, so the radius is.Solution 1We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largest is , and the middle one is , so the radius is .Solution 224. Squares, , andare equal in area. Pointsandare the midpointsof sidesand, respectively. What is the ratio of the area of the shaded pentagonto the sum of the areas of the three squares?ProblemSolution 1First let(whereis the side length of the squares) for simplicity. We can extenduntil it hits the extension of. Call this point. The area of trianglethen isThe area of rectangleis. Thus, our desired area is. Now, the ratio of the shaded area to the combined area of the three squares is.Solution 2Let the side length of each square be 1.Let the intersection ofandbe .Since, . Sinceand are vertical angles, theyare congruent. We also haveby definition.So we haveby congruence. Therefore,.Since andare midpoints of sides,. This combined withyields.The area of trapezoidis.The area of triangleis.So the area of the pentagon is .The area of the 3 squares is . Therefore, .Solution 3Let the intersection of andbe .Now we haveand .Because both triangles has a side on congruent squares therefore.Becauseand are vertical angles. Also bothand are right angles so .Therefore by AAS (Angle, Angle, Side) . Then translating/rotating the shadedinto the position ofSo the shaded area now completely covers the squareSet the area of a square asTherefore, .25.A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are inches, inches, andinches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?ProblemThe radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball (see figure below), you see that in A and C it loses inches, and it gains inches on B.So, the departurefrom the length of the track means that the answer is .Solution 1The total length of all of the arcs is . Since we want the path fromthe center, the actual distance will be shorter. Therefore, the only answer choice less thanis . This solution may be invalid because the actual distance can be longer if the path the center travels is on the outside of the curve, as it is in the middle bump. Solution 2。

2010_2015年AMC10A和B竞赛真题与答案(英文版)

2013年AMC 10A竞赛真题及答案（英文版）
201052015年aamc10a和bb竞赛真题及答案英文版你好目前只分享了20102013年amc10a和b竞赛真题及答案英文版20142015年的暂时还没有上传等采纳后再私信我吧
2010-2015年AMC 10A和B竞赛真题及答案（英文版）
你好，目前，只分享了2010-2013年AMC 10A和B竞赛真题及答案（英文版），2014-2015年的暂时还没有上传，等采纳后再私信我吧！需要的话，我可以将Word文档中的原图文件一同发给你，求采纳，毕竟我花了差不多一个下午才整理完，谢谢！(@_@)
2010年AMC 10A竞赛真题及答案（英文版）
2010年AMC 10B竞赛真题及答赛真题及答案（英文版）
2011年AMC 10B竞赛真题及答案（英文版）
Problem 4
2012年AMC 10A竞赛真题及答案（英文版）
2012年AMC 10B竞赛真题及答案（英文版）

2013年最新美国数学竞赛试题及答案(AMC2013A)

A1Square has side length . Point is on , and the area of is . What is ?2A softball team played ten games, scoring , and runs. They lost by one run inexactly five gam es. In each of the other gam es, they scored twice as m any runs as their opponent.How many total runs did their opponents score?3 A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of thepink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?4What is the value of5Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their tripTom paid , Dorothy paid , and Sammy paid . In order to share the costs equally, Tom gave Sammy dollars, and Dorothy gave Sammy dollars. What is ?6In a recent basketball game, Shenille attem pted only three-point shots and two-point shots. She wassuccessful on of her three-point shots and of her two-point shots. Shenille attem pted shots. How m any points did she score?7The sequence has the property that every term beginning with the third is thesum of the previous two. That is, Suppose that and. What is ?8Given that and are distinct nonzero real numbers such that , what is ?9In , and . Points and are on sides , , and ,respectively, such that and are parallel to and , respectively. What is the perimeter of parallelogram?(9th)（11 th）10Let be the set of positive integers for which has the repeating decimal representationwith and different digits. What is the sum of the elem ents of ?11Triangle is equilateral with . Points and are on and points and are onsuch that both and are parallel to . Furthermore, triangle and trapezoidsand all have the sam e perimeter. What is ?12he angles in a particular triangle are in arithmetic progression, and the side lengths are . Thesum of the possible values of equals where , and are positive integers. What is ?13Let points and . Quadrilateral is cut intoequal area pieces by a line passing through . This line intersects at point , where thesefractions are in lowest term s. What is ?14The sequence, , , ,is an arithm etic progression. What is ?15Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?16, , are three piles of rocks. The m ean weight of the rocks in is pounds, the m ean weightof the rocks in is pounds, the m ean weight of the rocks in the com bined piles and ispounds, and the m ean weight of the rocks in the combined piles and is pounds. What is thegreatest possible integer value for the mean in pounds of the rocks in the com bined piles and ?17A group of pirates agree to divide a treasure chest of gold coins am ong them selves as follows. Thepirate to take a share takes of the coins that rem ain in the chest. The number of coins initially in the chest is the sm allest number for which this arrangement will allow each pirate to receive apositive whole number of coins. How many coins doe the pirate receive?18Six spheres of radius are positioned so that their centers are at the vertices of a regular hexagon of side length . The six spheres are internally tangent to a larger sphere whose center is the center ofthe hexagon. An eighth sphere is externally tangent to the six sm aller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?19In , , and . A circle with center and radius intersects at pointsand . Moreover and have integer lengths. What is ?20Let be the set . For , define to m ean that either or. How m any ordered triples of elem ents of have the property that ,, and ?21Consider . Which of the following intervals contains ?22A palindrome is a nonnegatvie integer number that reads t he sam e forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome is chosen uniformly at random. Whatis the probability that is also a palindrome?23is a square of side length . Point is on such that . The square regionbounded by is rotated counterclockwise with center , sweeping out a region whosearea is , where , , and are positive integers and . What is ?24Three distinct segm ents are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segm ents are the three side lengths of a triangle with positive area?25Let be defined by . How m any complex numbers are there suchthat and both the real and the imaginary parts of are integers with absolute value atmost ?1. E2. C3. E4. C5. B6. B7. C8. D9. C10. D11. C12. A13. B14. B15. D16. E17. D18. B19. D20. B21. A22. E23. C24. E25. A。

amc 美国数学竞赛 00 amc 0b 试题及答案解析

2003 AMC 10B1、Which of the following is the same asSolution2、Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs more than a pinkpill, and Al’s pills cost a total of for the two weeks. How much doesone green pill cost?Solution3、The sum of 5 consecutive even integers is less than the sum of theﬁrst consecutive odd counting numbers. What is the smallest of theeven integers?Solution4、Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the ﬁgure. She plants one flower per square foot in each region. Asters cost 1 each,begonias 1.50 each, cannas 2 each, dahlias 2.50 each, and Easterlilies 3 each. What is the least possible cost, in dollars, for hergarden?Solution5、Moe uses a mower to cut his rectangular -foot by -foot lawn.The swath he cuts is inches wide, but he overlaps each cut byinches to make sure that no grass is missed. He walks at the rate of feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn?Solution.6、Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is . The horizontal length ofa “-inch” television screen is closest, in inches, to which of thefollowing?Solution7、The symbolism denotes the largest integer not exceeding . Forexample. , and . ComputeSolution.8、The second and fourth terms of a geometric sequence are and .Which of the following is a possible first term?Solution9、Find the value of that satisfies the equationSolution10、Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times is the number of possible license plates increased?Solution11、A line with slope intersects a line with slope at the point .What is the distance between the -intercepts of these two lines?Solution12、Al, Betty, and Clare split among them to be invested indifferent ways. Each begins with a different amount. At the end of one year they have a total of . Betty and Clare have both doubledtheir money, whereas Al has managed to lose . What was Al’soriginal portion?Solution.13、Let denote the sum of the digits of the positive integer . Forexample, and . For how many two-digitvalues of is ?Solution14、Given that , where both and are positive integers,find the smallest possible value for .Solution15、There are players in a singles tennis tournament. Thetournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest players aregiven a bye, and the remaining players are paired off to play. Aftereach round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played isSolution16、A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year ?Solution.17、An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly ﬁll the cone. Assume that the melted ice cream occupies of the volume of the frozen ice cream. What is theratio of the cone’s height to its radius?Solution18、What is the largest integer that is a divisor offor all positive even integers ?Solution19、Three semicircles of radius are constructed on diameter of asemicircle of radius . The centers of the small semicircles divideinto four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside thesmaller semicircles?Solution20、In rectangle , and . Points and are onso that and . Lines and intersect at . Findthe area of .Solution21、A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements?22、A clock chimes once at minutes past each hour and chimes onthe hour according to the hour. For example, at 1 PM there is one chime and at noon and midnight there are twelve chimes. Starting at11:15 AM on February , , on what date will the chimeoccur?Solution23、A regular octagon has an area of one square unit.What is the area of the rectangle ?Solution24、The ﬁrst four terms in an arithmetic sequence are , , ,and , in that order. What is the ﬁfth term?Solution25、How many distinct four-digit numbers are divisible by and haveas their last two digits?Solution。

-年amc 10a和b竞赛真题及答案(英文版)

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2010-2015年AMC 10A和B竞赛真题及答案（英文版）
你好，目前，只分享了2010-2013年AMC 10A和B竞赛真题及答案（英文版），2014-2015年的暂时还没有上传，等采纳后再私信我吧！需要的话，我可以将Word文档中的原图文件一同发给你，求采纳，毕竟我花了差不多一个下午才整理完，谢谢！(@_@)
2010年AMC 10Aห้องสมุดไป่ตู้赛真题及答案（英文版）
2010年AMC 10B竞赛真题及答案（英文版）
2011年AMC 10A竞赛真题及答案（英文版）
2011年AMC 10B竞赛真题及答案（英文版）
Problem 4
2012年AMC 10A竞赛真题及答案（英文版）
go去wentgone
2012年AMC 10B竞赛真题及答案（英文版）
get得到gotgot
sink下沉sank / sunksunk / sunken
give给gavegiven
hit打hithit
不规则动词表
2013年AMC 10A竞赛真题及答案（英文版）
mistake误认mistookmistaken
grow成长grewgrown
think思考thoughtthought

美国数学竞赛往年试题及答案

美国数学竞赛往年试题及答案试题一：一个圆的半径为 $ r $ 厘米，求圆的周长。

答案：圆的周长 $ C $ 可以用公式 $ C = 2\pi r $ 计算，其中$ \pi $ 是圆周率，大约等于3.14。

试题二：如果一个直角三角形的两条直角边分别为3厘米和4厘米，求斜边的长度。

答案：根据勾股定理，斜边 $ c $ 的长度可以通过公式 $ c =\sqrt{a^2 + b^2} $ 计算，其中 $ a $ 和 $ b $ 是直角边的长度。

所以斜边的长度为 $ c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $ 厘米。

试题三：一个数列的前三项是1, 1, 2。

从第四项开始，每一项都是前三项的和。

求第10项的值。

答案：数列的前几项为：1, 1, 2, 4, 7, 13, 24, 44, 81, 149。

第10项的值为149。

试题四：在一个不透明的袋子里，有3个红球和2个蓝球。

随机抽取2个球，求至少抽到1个红球的概率。

答案：总共有 $ \binom{5}{2} = 10 $ 种抽取2个球的组合。

至少抽到1个红球的组合有 $ \binom{3}{1} \times \binom{2}{1} +\binom{3}{2} = 6 + 3 = 9 $ 种。

所以概率为 $ \frac{9}{10} $。

试题五：一个函数 $ f(x) = x^2 - 6x + 8 $，求其顶点坐标。

答案：顶点的 $ x $ 坐标可以通过公式 $ x = -\frac{b}{2a} $ 计算，其中 $ a = 1 $ 和 $ b = -6 $。

所以 $ x = -\frac{-6}{2\times 1} = 3 $。

将 $ x = 3 $ 代入函数得到 $ y = 3^2 - 6\times 3 + 8 = -1 $。

美国数学竞赛AMC题目及答案

2.3.What is the value of ?4.Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?5.Hammie is in the grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?6.The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?7.Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?8.A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?9.The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?10.What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?11. Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?12. At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?13. When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?14. Let the two digits be and .The correct score was . Clara misinterpreted it as . The difference between the two is which factors into . Therefore, since the difference is a multiple of 9, the only answerchoice that is a multiple of 9 is .15. If , , and , what is the product of , , and ?16. A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of -graders to -graders is , and the the ratio of -graders to -graders is . What is the smallest number of students that could be participating in the project?17. The sum of six consecutive positive integers is 2013. What is the largest of these six integers?18. Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?--Arpanliku 16:22, 27 November 2013 (EST) Courtesy of Lord.of.AMC19. Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?20. A rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?21. Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?22. Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?23. Angle of is a right angle. The sides of are the diameters of semicircles as shown. The area of the semicircle on equals , and the arc of the semicircle on has length .What is the radius of the semicircle on ?24. Squares , , and are equal in area. Points and are the midpoints of sidesand , respectively. What is the ratio of the area of the shaded pentagon to the sum of the areas of the three squares?25. A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are inches, inches, and inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of theball travels over the course from A to B?1.2.The 50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound of fish is $6. Consequently, if half a pound of fish costs $6, then a whole pound of fish is dollars.3.Notice that we can pair up every two numbers to make a sum of 1:Therefore, the answer is .4.Each of her seven friends paid to cover Judi's portion. Therefore, Judi's portion must be . Since Judi was supposed to pay of the total bill, the total bill must be .5.The median here is obviously less than the mean, so option (A) and (B) are out.Lining up the numbers (5, 5, 6, 8, 106), we see that the median weight is 6 pounds.The average weight of the five kids is .Therefore, the average weight is bigger, by pounds, making the answer.6.Solution 1: Working BackwardsLet the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.We see that , making .It follows that , so .Solution 2: Jumping Back to the StartAnother way to do this problem is to realize what makes up the bottommost number. This method doesn't work quite as well for this problem, but in a larger tree, it might be faster. (In this case, Solution 1 would be faster since there's only two missing numbers.)Again, let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.We can write some equations:Now we can substitute into the first equation using the two others:7.If Trey saw , then he saw .2 minutes and 45 seconds can also be expressed as seconds.Trey's rate of seeing cars, , can be multiplied by on the top and bottom (and preserve the same rate):. It follows that the most likely number of cars is . Solution 2minutes and seconds is equal to .Since Trey probably counts around cars every seconds, there are groups of cars that Trey most likely counts. Since , the closest answer choice is .8.First, there are ways to flip the coins, in order.The ways to get two consecutive heads are HHT and THH.The way to get three consecutive heads is HHH.Therefore, the probability of flipping at least two consecutive heads is .9.This is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters, we note that .However, because the first term is and not , the solution to the problem is10. To find either the LCM or the GCF of two numbers, always prime factorize first.The prime factorization of .The prime factorization of .Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is ). Multiply all of these to get 5940.For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply. = 18.Thus the answer = = .We start off with a similar approach as the original solution. From the prime factorizations, the GCF is .It is a well known fact that . So we have,.Dividing by yields .Therefore, .11. We use that fact that . Let d= distance, r= rate or speed, and t=time. In this case, let represent the time.On Monday, he was at a rate of . So, .For Wednesday, he walked at a rate of . Therefore, .On Friday, he walked at a rate of . So, .Adding up the hours yields + + = .We now find the amount of time Grandfather would have taken if he walked at per day. Set up the equation, .To find the amount of time saved, subtract the two amounts: - = . To convert this to minutes, we multiply by .Thus, the solution to this problem is12. First, find the amount of money one will pay for three sandals without the discount. We have.Then, find the amount of money using the discount: .Finding the percentage yields .To find the percent saved, we have13. Let the two digits be and .The correct score was . Clara misinterpreted it as . The difference between the two is which factors into . Therefore, since the difference is a multiple of 9, the only answerchoice that is a multiple of 9 is .14. The probability that both show a green bean is . The probability that both show a red bean is . Therefore the probability is15.Therefore, .Therefore, .To most people, it would not be immediately evident that , so we can multiply 6's until we get the desired number:, so .Therefore the answer is .16. Solution 1: AlgebraWe multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:Therefore, the ratio of 8th graders to 7th graders to 6th graders is . Since the ratio is in lowest terms, the smallest number of students participating in the project is .Solution 2: FakesolvingThe number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallest possibility). Then there are 6th graders and 7th graders. The numbers of students is17. Solution 1The mean of these numbers is . Therefore the numbers are, so the answer isSolution 2Let the number be . Then our desired number is .Our integers are , so we have that.Solution 3Let the first term be . Our integers are . We have,18. Solution 1There are cubes on the base of the box. Then, for each of the 4 layers above the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there arecubes. Hence, the answer is .Solution 2We can just calculate the volume of the prism that was cut out of the original box. Each interior side of the fort will be feet shorter than each side of the outside. Since the floor is foot, the height will be feet. So the volume of the interior box is .The volume of the original box is . Therefore, the number of blocks contained inthe fort is .19. If Hannah did better than Cassie, there would be no way she could know for sure that she didn't get the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, if Hannah did worse than Bridget, there is no way Bridget could have known that she didn't get the highest in the class.Therefore, Hannah did better than Bridget, so our order is .20.A semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem, . The area is .21.The number of ways to get from Samantha's house to City Park is , and the number of ways toget from City Park to school is . Since there's one way to go through City Park (just walking straight through), the number of different ways to go from Samantha's house to City Park to school .22. There are vertical columns with a length of toothpicks, and there are horizontal rows with a length of toothpicks. An effective way to verify this is to try a small case, i.e. a grid of toothpicks.Thus, our answer is .23. Solution 1If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagoreantheorem says that the other side has length 15, so the radius is .Solution 2We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of thelargest is , and the middle one is , so the radius is .24.First let (where is the side length of the squares) for simplicity. We can extend until it hits theextension of . Call this point . The area of triangle then is The area of rectangle is . Thus, our desired area is . Now, the ratio of the shaded area to thecombined area of the three squares is .Let the side length of each square be .Let the intersection of and be .Since , . Since and are vertical angles, they are congruent. We also have by definition.So we have by congruence. Therefore, .Since and are midpoints of sides, . This combined with yields.The area of trapezoid is .The area of triangle is .So the area of the pentagon is .The area of the squares is .Therefore, .Let the intersection of and be .Now we have and .Because both triangles has a side on congruent squares therefore .Because and are vertical angles .Also both and are right angles so .Therefore by AAS(Angle, Angle, Side) .Then translating/rotating the shaded into the position ofSo the shaded area now completely covers the squareSet the area of a square asTherefore, .25. Solution 1The radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball (see figure below), you see that in A and C it loses inches, and it gains inches on B.So, the departure from the length of the track means that the answer is . Solution 2The total length of all of the arcs is . Since we want the path from the center, the actual distance will be shorter. Therefore, the only answer choice less than is . Thissolution may be invalid because the actual distance can be longer if the path the center travels is on the outside of the curve, as it is in the middle bump.古希腊哲学大师亚里士多德说：人有两种，一种即“吃饭是为了活着”,一种是“活着是为了吃饭”.一个人之所以伟大，首先是因为他有超于常人的心。

AMC美国数学竞赛AMC.B 试题及答案解析

2003A M C10 B 1、Which of the following is the same as2、Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs more than a pink pill, and Al’s pills cost a total of for the two weeks. How much does one green pill cost?3、The sum of 5 consecutive even integers is less than the sum of the ?rst consecutive odd counting numbers. What is the smallest of the even integers?4、Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the ?gure. She plants one flower per square foot in each region. Asters cost 1 each, begonias each, cannas 2 each, dahlias each, and Easter lilies 3 each. What is the least possible cost, in dollars, for her garden?5、Moe uses a mower to cut his rectangular -foot by -foot lawn. The swath he cuts is inches wide, but he overlaps each cut by inches tomake sure that no grass is missed. He walks at the rate of feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn?.6、Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is . The horizontal length of a “-inch” television screen is closest, in inches, to which of the following?7、The symbolism denotes the largest integer not exceeding . For example. , and . Compute.8、The second and fourth terms of a geometric sequence are and . Which of the following is a possible first term?9、Find the value of that satisfies the equation10、Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times is the number of possible license plates increased?11、A line with slope intersects a line with slope at the point . What is the distance between the -intercepts of these two lines?12、Al, Betty, and Clare split among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of . Betty and Clare have both doubled their money, whereas Al has managed to lose . What was Al’s origin al portion?.13、Let denote the sum of the digits of the positive integer . For example, and . For how many two-digit values of is ?14、Given that , where both and are positive integers, find the smallest possible value for .15、There are players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest players are given a bye, and the remaining players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is16、A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year ?.17、An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly ?ll the cone. Assume that the melted ice cream occupies of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?18、What is the largest integer that is a divisor offor all positive even integers ?19、Three semicircles of radius are constructed on diameter of a semicircle of radius . The centers of the small semicircles divide into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?20、In rectangle , and . Points and are on so that and . Lines and intersect at . Find the area of .21、A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements?22、A clock chimes once at minutes past each hour and chimes on the hour according to the hour. For example, at 1 PM there is one chime and at noon and midnight there are twelve chimes. Starting at 11:15 AM on February , , on what date will the chime occur?23、A regular octagon has an area of one square unit. What is the area of the rectangle ?24、The ?rst four terms in an arithmetic sequence are , , , and, in that order. What is the ?fth term?25、How many distinct four-digit numbers are divisible by and have as their last two digits?。