Competition · AMC preparation · step 4 of 4

AMC 10 2015A: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 10 2015A #1 grade 8+ arithmetic
What is the value of (2⁰-1+5²-0)⁻¹×5?
AMC 10 2015A #2 grade 7+ geometry-2d
A box contains a collection of triangular and square tiles. There are 25 tiles in the box, containing 84 edges total. Ho…
AMC 10 2015A #3 grade 4+ arithmetic
Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to comp…
AMC 10 2015A #4 grade 6+ rate-ratio
Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice a…
AMC 10 2015A #5 grade 6+ arithmetic
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payto…
AMC 10 2015A #6 grade 7+ rate-ratio
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller numbe…
AMC 10 2015A #7 grade 4+ arithmetic
How many terms are in the arithmetic sequence 13, 16, 19, …, 70, 73?
AMC 10 2015A #8 grade 8+ rate-ratio
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Cla…
AMC 10 2015A #9 grade 8+ geometry-3d
Two right circular cylinders have the same volume. The radius of the second cylinder is 10% more than the radius of the…
AMC 10 2015A #10 grade 2+ arithmetic
How many rearrangements of abcd are there in which no two adjacent letters are also adjacent letters in the alphabet? Fo…
AMC 10 2015A #11 grade 8+ geometry-2d
The ratio of the length to the width of a rectangle is 4 : 3. If the rectangle has diagonal of length d, then the a…
AMC 10 2015A #12 grade 8+ arithmetic
Points ( √π , a) and ( √π , b) are distinct points on the graph of y² + x⁴ = 2x² y + 1. What is |a-b|?
AMC 10 2015A #13 grade 6+ arithmetic
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 b…
AMC 10 2015A #14 grade 7+ geometry-2d
The diagram below shows the circular face of a clock with radius 20 cm and a circular disk with radius 10 cm externally…
AMC 10 2015A #15 grade 7+ rate-ratio
Consider the set of all fractions x/y, where x and y are relatively prime positive integers. How many of these fractions…
AMC 10 2015A #16 grade 8+ arithmetic
If y+4 = (x-2)², x+4 = (y-2)², and x ≠ y, what is the value of x²+y²?
AMC 10 2015A #17 grade 8+ geometry-2d
A line that passes through the origin intersects both the line x = 1 and the line y=1+ √3/3 x. The three lines create an…
AMC 10 2015A #18 grade 7+ arithmetic
Hexadecimal (base-16) numbers are written using numeric digits 0 through 9 as well as the letters A through F to represe…
AMC 10 2015A #19 grade 8+ geometry-2d
The isosceles right triangle ABC has right angle at C and area 12.5. The rays trisecting ∠ ACB intersect AB at D and E.…
AMC 10 2015A #20 grade 6+ geometry-2d
A rectangle with positive integer side lengths in cm has area A cm² and perimeter P cm. Which of the following numbers c…
AMC 10 2015A #21 grade 8+ geometry-3d
Tetrahedron ABCD has AB=5, AC=3, BC=4, BD=4, AD=3, and CD=12/5√2. What is the volume of the tetrahedron?
AMC 10 2015A #22 grade 7+ probability
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those…
AMC 10 2015A #23 grade 8+ arithmetic
The zeroes of the function f(x)=x²-ax+2a are integers. What is the sum of the possible values of a?
AMC 10 2015A #24 grade 8+ geometry-2d
For some positive integers p, there is a quadrilateral ABCD with positive integer side lengths, perimeter p, right angle…
AMC 10 2015A #25 grade 8+ geometry-2d
Let S be a square of side length 1. Two points are chosen independently at random on the sides of S. The probability tha…

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