Competition · AMC preparation · step 4 of 4

AMC 12 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 12 2015A #1 grade 8+ arithmetic
What is the value of (2⁰-1+5²-0)⁻¹×5?
AMC 12 2015A #2 grade 7+ geometry-2d
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the tr…
AMC 12 2015A #3 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 12 2015A #4 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 12 2015A #5 grade 7+ logic, arithmetic
Amelia needs to estimate the quantity a/b - c, where a, b, and c are large positive integers. She rounds each of the int…
AMC 12 2015A #6 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 12 2015A #7 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 12 2015A #8 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 12 2015A #9 grade 7+ probability
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then…
AMC 12 2015A #10 grade 7+ algebra, number-theory
Integers x and y with x>y>0 satisfy x+y+xy=80. What is x?
AMC 12 2015A #11 grade 10+ geometry-2d
On a sheet of paper, Isabella draws a circle of radius 2, a circle of radius 3, and all possible lines simultaneously ta…
AMC 12 2015A #12 grade 9+ geometry-2d
The parabolas y=ax² - 2 and y=4 - bx² intersect the coordinate axes in exactly four points, and these four points are th…
AMC 12 2015A #13 grade 6+ logic, counting
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games eithe…
AMC 12 2015A #14 grade 11+ algebra
What is the value of a for which 1/log₂a + 1/log₃a + 1/log₄a = 1?
AMC 12 2015A #15 grade 8+ rate-ratio
What is the minimum number of digits to the right of the decimal point needed to express the fraction 123456789/(2²⁶· 5⁴…
AMC 12 2015A #16 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 12 2015A #17 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 12 2015A #18 grade 8+ algebra, number-theory
The zeroes of the function f(x)=x²-ax+2a are integers. What is the sum of the possible values of a?
AMC 12 2015A #19 grade 8+ geometry-2d
For some positive integers p, there is a quadrilateral ABCD with positive integer side lengths, perimeter p, right angle…
AMC 12 2015A #20 grade 11+ geometry-2d
Isosceles triangles T and T' are not congruent but have the same area and the same perimeter. The sides of T have length…
AMC 12 2015A #21 grade 11+ geometry-2d
A circle of radius r passes through both foci of, and exactly four points on, the ellipse with equation x²+16y²=16. The…
AMC 12 2015A #22 grade 9+ counting, pattern
For each positive integer n, let S(n) be the number of sequences of length n consisting solely of the letters A and B, w…
AMC 12 2015A #23 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…
AMC 12 2015A #24 grade 11+ probability
Rational numbers a and b are chosen at random among all rational numbers in the interval [0,2) that can be written as fr…
AMC 12 2015A #25 grade 11+ geometry-2d
A collection of circles in the upper half-plane, all tangent to the x-axis, is constructed in layers as follows. Layer L…

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