Competition · AMC preparation · step 4 of 4

AMC 12 2015B: 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 2015B #1 grade 8+ arithmetic
What is the value of 2-(-2)⁻²?
AMC 12 2015B #2 grade 4+ rate-ratio
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and f…
AMC 12 2015B #3 grade 4+ algebra
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and on…
AMC 12 2015B #4 grade 2+ logic
David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of…
AMC 12 2015B #5 grade 7+ rate-ratio
The Tigers beat the Sharks 2 out of the 3 times they played. They then played N more times, and the Sharks ended up winn…
AMC 12 2015B #6 grade 5+ rate-ratio
Back in 1930, Tillie had to memorize her multiplication facts from 0 × 0 to 12 × 12. The multiplication table she was gi…
AMC 12 2015B #7 grade 8+ geometry-2d
A regular 15-gon has L lines of symmetry, and the smallest positive angle for which it has rotational symmetry is R degr…
AMC 12 2015B #8 grade 11+ algebra
What is the value of (625^(log₅ 2015))^(1/4) ?
AMC 12 2015B #9 grade 11+ probability
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. Th…
AMC 12 2015B #10 grade 8+ geometry-2d
How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, iso…
AMC 12 2015B #11 grade 8+ geometry-2d
The line 12x+5y=60 forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this tr…
AMC 12 2015B #12 grade 8+ algebra
Let a, b, and c be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation (…
AMC 12 2015B #13 grade 10+ geometry-2d
Quadrilateral ABCD is inscribed in a circle with ∠ BAC=70°, ∠ ADB=40°, AD=4, and BC=6. What is AC?
AMC 12 2015B #14 grade 8+ geometry-2d
A circle of radius 2 is centered at A. An equilateral triangle with side 4 has a vertex at A. What is the difference bet…
AMC 12 2015B #15 grade 7+ probability
At Rachelle's school, an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA in the four classes she…
AMC 12 2015B #16 grade 10+ geometry-3d
A regular hexagon with sides of length 6 has an isosceles triangle attached to each side. Each of these triangles has tw…
AMC 12 2015B #17 grade 11+ probability
An unfair coin lands on heads with a probability of 1/4. When tossed n>1 times, the probability of exactly two heads is…
AMC 12 2015B #18 grade 8+ number-theory
For every composite positive integer n, define r(n) to be the sum of the factors in the prime factorization of n. For ex…
AMC 12 2015B #19 grade 8+ geometry-2d
In △ABC, ∠C = 90° and AB = 12. Squares ABXY and ACWZ are constructed outside of the triangle. The points X, Y, Z, and W…
AMC 12 2015B #20 grade 8+ pattern, number-theory
For every positive integer n, let mod₅ (n) be the remainder obtained when n is divided by 5. Define a function f: {0,1,2…
AMC 12 2015B #21 grade 6+ number-theory
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up th…
AMC 12 2015B #22 grade 9+ counting
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits d…
AMC 12 2015B #23 grade 7+ geometry-3d
A rectangular box measures a × b × c, where a, b, and c are integers and 1≤ a ≤ b ≤ c. The volume and the surface area o…
AMC 12 2015B #24 grade 10+ geometry-2d
Four circles, no two of which are congruent, have centers at A, B, C, and D, and points P and Q lie on all four circles.…
AMC 12 2015B #25 grade 11+ geometry-2d, pattern
A bee starts flying from point P₀. She flies 1 inch due east to point P₁. For j ≥ 1, once the bee reaches point P_j, she…

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