2015 AMC 10A Problems

2015 AMC 10A Problems 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. 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 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?

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 What is

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?

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?

$ \\textbf{(A) }\\mathrm{2 o'clock} \\qquad\\textbf{(B) }\\mathrm{3 o'clock}

\\qquad\\textbf{(C) }\\mathrm{4 o'clock} \\qquad\\textbf{(D) }\\mathrm{6 o'clock} \\qquad\\textbf{(E) }\\mathrm{8 o'clock} $ Problem 15

Consider the set of all fractions

where and are relatively prime positive integers.

?

Problem 16 If

?

Problem 17

, and

, what is the value of

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

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

. 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 trisecting intersect

has right angle at and area

at and . What is the area of

. The rays

?

Problem 20

A rectangle with positive integer side lengths in Which of the following numbers cannot equal

has area ?

and perimeter

.

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 Tetrahedron

has

,

,

,

,

, and

. What is the volume of the tetrahedron?

Problem 22

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?

Problem 23

The zeros of the function possible values of ?

Problem 24

For some positive integers , there is a quadrilateral lengths, perimeter , right angles at and , different values of

are possible?

Problem 25

Let be a square of side length . Two points are chosen independently at random on the sides of . The probability that the straight-line distance between the points is at least is

, where , , and are positive integers with ?

Answer key

Abcde abcde abcde abcde abcde

. What is

with positive integer side , and . How many are integers. What is the sum of the

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