2015 AMC 10A 考题及答案

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2015 AMC 10A 考题及答案

Problem 1 What is the value of

Problem 2

A box contains a collection of triangular and square tiles. There are tiles in the box, containing edges total. How many square tiles are there in the box?

Problem 3

Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase?

Problem 4

Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three

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will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?

Problem 5

Mr. Patrick teaches math to students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was . After he graded Payton's test, the test average became . What was Payton's score on the test?

Problem 6

The sum of two positive numbers is times their difference. What is the ratio of the larger number to the smaller number?

Problem 7

How many terms are there in the arithmetic sequence , ?

Problem 8

Two years ago Pete was three times as old as his cousin Claire. Two

years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be : ?

Problem 9

Two right circular cylinders have the same volume. The radius of the second cylinder is more than the radius of the first. What is the relationship between the heights of the two cylinders?

,

,

, . . .,

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Problem 10

How many rearrangements of are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either or .

Problem 11

The ratio of the length to the width of a rectangle is : . If the rectangle has diagonal of length , then the area may be expressed as for some constant . What is ?

Problem 12 Points

and

are distinct points on the graph of

?

Problem 13

Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as

combinations of one or more of her coins. How many 10-cent coins does Claudia have?

. What is

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Problem 14

The diagram below shows the circular face of a clock with radius cm and a circular disk with radius cm externally tangent to the clock face at o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?

Problem 15

Consider the set of all fractions

where and are relatively prime

positive integers. How many of these fractions have the property that if both numerator and denominator are increased by , the value of the fraction is increased by ?

Problem 16 If

?

, and

, what is the value of

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Problem 17

A line that passes through the origin intersects both the line the line

and

. The three lines create an equilateral triangle.

What is the perimeter of the triangle?

Problem 18

Hexadecimal (base-16) numbers are written using numeric digits through as well as the letters through to represent through . Among the first positive integers, there are whose

hexadecimal representation contains only numeric digits. What is the sum of the digits of ?

Problem 19

The isosceles right triangle has right angle at and area . The rays trisecting intersect at and . What is the area of ?

Problem 20

A rectangle with positive integer side lengths in has area and perimeter . Which of the following numbers cannot equal

?

NOTE: As it originally appeared in the AMC 10, this problem was stated incorrectly and had no answer; it has been modified here to be solvable.

Problem 21

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