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2013 年美国数学竞赛 AMC10B 真题

Problem 1

What is ?

Problem 2

Mr. Green measures his rectangular garden by walking two of the sides and finding that it is by steps. Each of Mr. Green's steps is steps

feet long. Mr. Green expects a half a pound of potatoes per

square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?

Problem 3

On a particular January day, the high temperature in Lincoln, Nebraska, was degrees higher than . In degrees, what

the low temperature, and the average of the high and the low temperatures was was the low temperature in Lincoln that day?

Problem 4

When counting from from to , to , is the number counted. When counting backwards ?

is the

number counted. What is

Problem 5

Positive integers and are each less than . What is the smallest possible value for ?

1

Problem 6

The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders?

Problem 7

Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?

Problem 8

Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?

Problem 9

Three positive integers are each greater than , have a product of prime. What is their sum? , and are pairwise relatively

Problem 10

A basketball team's players were successful on 50% of their two-point shots and 40% of their threepoint shots, which resulted in 54 points. They attempted 50% more two-point shots than three-point shots. How many three-point shots did they attempt?

2

Problem 11

Real numbers and satisfy the equation . What is ?

Problem 12

Let be the set of sides and diagonals of a regular pentagon. A pair of elements of are selected at

random without replacement. What is the probability that the two chosen segments have the same length?

Problem 13

Jo and Blair take turns counting from to one more than the last number said by the other person. Jo " . Jo then says " " , and so on. What is the

starts by saying " ", so Blair follows by saying " 53rd number said?

Problem 14

Define which ? . Which of the following describes the set of points for

Problem 15

A wire is cut into two pieces, one of length and the other of length . The piece of length is bent to

form an equilateral triangle, and the piece of length and the hexagon have equal area. What is ?

is bent to form a regular hexagon. The triangle

3

Problem 16

In triangle the area of , medians ? and intersect at , , , and . What is

Problem 17

Alex has red tokens and blue tokens. There is a booth where Alex can give two red tokens and

receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

Problem 18

The number is has the property that its units digit is the sum of its other digits, that . How many integers less than but greater than share this property?

Problem 19

The real numbers quadratic form an arithmetic sequence with has exactly one root. What is this root? . The

Problem 20

The number where possible. What is ? is expressed in the form and are positive integers and is as small as

4

Problem 21

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is . What is the smallest possible value of N?

Problem 22

The regular octagon associated with one of the digits the numbers on the lines done? , has its center at . Each of the vertices and the center are to be

through , with each digit used once, in such a way that the sums of , , and are all equal. In how many ways can this be

Problem 23

In triangle segments , , , and , , and , respectively, such that , where and . Distinct points , , , and , and lie on . The

length of segment What is ?

can be written as

are relatively prime positive integers.

Problem 24

A positive integer (including set and is nice if there is a positive integer with exactly four positive divisors . How many numbers in the

) such that the sum of the four divisors is equal to are nice?

5

Problem 25

Bernardo chooses a three-digit positive integer and writes both its base-5 and base-6

representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer Bernardo writes the numbers many choices of and . For example, if , . For how ?

, and LeRoy obtains the sum , in order, the same as those of

are the two rightmost digits of

6

Answer Key

1. C 2. A 3. C 4. D 5. B 6. C 7. B 8. B 9. D 10. C 11. B 12. B 13. E 14. E 15. B 16. B 17. E 18. D 19. D 20. B 21. C 22. C 23. B 24. A 25. E

7

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