2010AMC10美国数学竞赛A卷

2010 AMC10美国数学竞赛A卷

1. Mary’s top book shelf holds five books with the following widths, in centimeters: 6, 1/2, 1,

2.5, and 10. What is the average book width, in centimeters?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

2. Four 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?

(A) 5/4 (B) 4/3 (C) 3/2

(D) 2 (E) 3

3. Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?

(A) 3 (B) 13 (C) 18 (D) 25 (E) 29

4. A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disc can hold up to 56 minutes of reading. Assume that 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?

(A) 50.2 (B) 51.5 (C) 52.4 (D) 53.8 (E) 55.2

5. The area of a circle whose circumference is 24πis kπ. What is the value of k?

(A) 6 (B) 12 (C) 24 (D) 36 (E) 144

6. For positive numbers x and y the operation Φ(x, y) is defined as 1

(,)

x y x

Φ=-.

What is (2,2,2)

ΦΦ（）?

(A) 2

3 (B) 1(C) 4

(D) 5

(E) 2

7. Crystal 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?

(A)

1(B) (C) (D) 2 (E)

8. Tony works 2 hours a day and is paid $0.50 per hour for each full year of his age. During a six month period Tony worked 50 days and earned $630. How old was Tony at the end of the six month period?

(A) 9 (B) 11 (C) 12 (D) 13 (E) 14

9. A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x+32 are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?

(A) 20 (B) 21 (C) 22 (D) 23 (E) 24

11. The length of the interval of solutions of the inequality 23a x b

≤+≤ is 10. What is

b-a? (A) 6 (B) 10 (C) 15 (D) 20 (E) 30

12. Logan 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? (A) 0.04 (B)

0.4

(C)

0.4

(D)

(E) 4

13. Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time t in hours that she drove before her stop? (A) 880100(

)250

t t +-=

(B)

80250

t =

(D) 90250t = (E)

880(

)100250

t t -+=

14. Triangle ABC has AB=2AC. Let D and E be on AB and BC respectively, such that

∠BAE=∠ACD. Let F be the intersection of segments AE and CD, and suppose that △CFE is equilateral. What is ∠ACB?

(A) 60°(B)75°(C)90°(D) 105° (E) 120°

15. In 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.”

Mike:”Of the four of us, at least two are toads.”

How many of these amphibians are frogs?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

16. Non degenerate △ABC has integer side lengths, BD is an angle bisector, AD=3, and DC=8. What is the smallest possible value of the perimeter?

(A) 30 (B) 33 (C) 35 (D) 36 (E) 37

17. A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut 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?

(A) 7 (B) 8 (C) 10 (D) 12 (E) 15

18. Bernardo randomly picks 3distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8} and 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?

(A) 47

72(B) 37

(D) 49

(E) 39

19. Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r. The area of △ACE is 70% of the area of the hexagon. What is the sum of all possible values of r?

(A)

3(B) 10

(E) 6

20. A fly trapped inside a cubical box with side length 1 meter decides to relieve 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?

(A)

4+(B) 2+(C) 2+

(D) +(E)

21. The polynomial 322010

++-has three positive integer zeros. What is the

x a x b x

smallest possible value of a?

(A) 78 (B) 88 (C) 98 (D) 108 (E) 118

22. Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices is the interior of the circle are created?

(A) 28(B) 56(C) 70(D) 84(E) 140

23. Each of 2010 boxes in a line contains a single red marble, and for 12010

≤≤, the

k th position also contains k 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 a red marble. Let P(n) be the probability that Isabella stops after

drawing exactly n marbles. What is the smallest value of n for which 1

P n≤?

()

2010 (A) 45 (B) 63 (C) 64 (D) 201 (E) 1005

24. The number obtained from the last two nonzero digits of 90! Is equal to n. What is

(A) 12 (B) 32 (C) 48 (D) 52 (E) 68

25. Jim starts with a positive integer n 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 n=55, then his sequence contains 5 numbers:

55-72= 6

6-22= 2

2-12= 1

1-12= 0

Let N be the smallest number for which Jim’s sequence has 8 numbers. What is the units digit of N?

(A) 1 (B) 3 (C) 5 (D) 7 (E) 9

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●2008年考试情况 AMC/AIME中国历程： 1983第1届AIME上海有76名同学获得参赛资格 1984年第2届AIME有110人获得参赛资格 1985年第3届AIME北京有118名同学获得参赛资格 1986年第4届AIME上海有154名同学获得参赛资格，我国首次参加IMO的上海向明中学吴思皓就是在第四届AIME中获得满分 1992年第10届AIME上海有一千多名同学获得参赛资格，其中格致中学潘毅明，交大附中张觉，上海中学葛建庆均获满分1993年第11届AIME上海有一千多名同学获得参赛资格，其中华东师大二附中高一王海栋，格致中学高二（女）黄静，市西中学高二张

亮，复旦附中高三韩志刚四人获得满分，前三名总分排名复旦附中41分，华东师大二附中41分，上海中学40分。北京地区参加2006年AMC的共有7所市重点学校的842名学生，有515名学生获得参加AIME资格，其中，清华附中有61名学生参加AMC，45名学生获得AIME资格，20名学生获得荣誉奖章据悉中国大陆以下地区可以报名参加考试：北京地区：中国数学会奥林匹克委员会负责组织实施长春地区、哈尔滨地区也有参加考试在华举办的美国人子弟学校也有参加考试广州地区：《数学奥林匹克报》负责组织实施。在中国大陆报名者就在中国大陆考试。考题采用英文版。 2009年AMC中国地区参赛学校一览表

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2011AMC10美国数学竞赛A卷时间：2021.03.03 创作：欧阳学 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) $30.00 2. 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) 15 3. 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 ?

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2018 AIME I Problems Problem 1 Let be the number of ordered pairs of integers with and such that the polynomial can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when is divided by . Problem 2 The number can be written in base as , can be written in base as , and can be written in base as , where . Find the base- representation of . Problem 3 Kathy has red cards and green cards. She shuffles the cards and lays out of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is , where and are relatively prime positive integers. Find . Problem 4 In and . Point lies strictly between and on and point lies strictly between and on so that . Then can be expressed in the form , where and are relatively prime positive integers. Find . Problem 5 For each ordered pair of real numbers satisfying there is a real number such that

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2011AMC10美国数学竞赛A卷附中文翻译和答案

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