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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 m

any 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 number to the smaller number? times their difference. What is the ratio of the larger

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 length , then the area may be expressed as : . If the rectangle has diagonal of . What is ?

for some constant

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.

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

Problem 17

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 ? are integers. What is the sum of the

Problem 24 For some positive integers , there is a quadrilateral lengths, perimeter , right angles at and , different values of are possible? with positive integer side , and . How many

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 . What is

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