Competition · AMC preparation · step 4 of 4

AMC 12 2022A: 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 2022A #1 grade 5+ arithmetic
What is the value of 3+1/(3+1/(3+1/3))?
AMC 12 2022A #2 grade 6+ algebra
The sum of three numbers is 96. The first number is 6 times the third number, and the third number is 40 less than the s…
AMC 12 2022A #3 grade 6+ geometry-2d
Five rectangles, A, B, C, D, and E, are arranged in a square as shown below. These rectangles have dimensions 1×6, 2×4,…
AMC 12 2022A #4 grade 6+ number-theory
The least common multiple of a positive integer n and 18 is 180, and the greatest common divisor of n and 45 is 15. What…
AMC 12 2022A #5 grade 6+ counting
The taxicab distance between points (x₁, y₁) and (x₂, y₂) in the coordinate plane is given by |x₁ - x₂| + |y₁ - y₂|. For…
AMC 12 2022A #6 grade 6+ arithmetic
A data set consists of 6 (not distinct) positive integers: 1, 7, 5, 2, 5, and X. The average (arithmetic mean) of the 6…
AMC 12 2022A #7 grade 7+ counting, geometry-2d
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, bl…
AMC 12 2022A #8 grade 8+ algebra
The infinite product ³√10 · ³√³√10 · ³√³√³√10 … evaluates to a real number. What is that number?
AMC 12 2022A #9 grade 6+ logic
On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: S…
AMC 12 2022A #10 grade 7+ counting
How many ways are there to split the integers 1 through 14 into 7 pairs such that in each pair, the greater number is at…
AMC 12 2022A #11 grade 11+ algebra
What is the product of all real numbers x such that the distance on the number line between log₆x and log₆9 is twice the…
AMC 12 2022A #12 grade 11+ geometry-3d
Let M be the midpoint of AB in regular tetrahedron ABCD. What is cos(∠ CMD)?
AMC 12 2022A #13 grade 11+ geometry-2d
Let R be the region in the complex plane consisting of all complex numbers z that can be written as the sum of complex n…
AMC 12 2022A #14 grade 11+ algebra
What is the value of (log 5)³+(log 20)³+(log 8)(log 0.25) where log denotes the base-ten logarithm?
AMC 12 2022A #15 grade 8+ geometry-3d
The roots of the polynomial 10x³ - 39x² + 29x - 6 are the height, length, and width of a rectangular box (right rectangu…
AMC 12 2022A #16 grade 9+ number-theory
A triangular number is a positive integer that can be expressed in the form t_n=1+2+3+…+n, for some positive integer n.…
AMC 12 2022A #17 grade 11+ algebra
Suppose a is a real number such that the equation a·(sinx+sin(2x)) = sin(3x) has more than one solution in the interval…
AMC 12 2022A #18 grade 8+ geometry-2d
Let T_k be the transformation of the coordinate plane that first rotates the plane k degrees counterclockwise around the…
AMC 12 2022A #19 grade 8+ counting
Suppose that 13 cards numbered 1, 2, 3, …, 13 are arranged in a row. The task is to pick them up in numerically increasi…
AMC 12 2022A #20 grade 8+ geometry-2d
Isosceles trapezoid ABCD has parallel sides AD and BC, with BC < AD and AB = CD. There is a point P in the plane such th…
AMC 12 2022A #21 grade 9+ algebra
Let P(x) = x²⁰²² + x¹⁰¹¹ + 1. Which of the following polynomials is a factor of P(x)?
AMC 12 2022A #22 grade 11+ algebra, geometry-2d
Let c be a real number, and let z₁ and z₂ be the two complex numbers satisfying the equation z² - cz + 10 = 0. Points z₁…
AMC 12 2022A #23 grade 11+ number-theory
Let h_n and k_n be the unique relatively prime positive integers such that 1/1+1/2+1/3+…+1/n=(h_n)/(k_n). Let L_n denote…
AMC 12 2022A #24 grade 6+ counting
How many strings of length 5 formed from the digits 0, 1, 2, 3, 4 are there such that for each j ∈ {1,2,3,4}, at least j…
AMC 12 2022A #25 grade 10+ geometry-2d, number-theory
A circle with integer radius r is centered at (r, r). Distinct line segments of length c_i connect points (0, a_i) to (b…

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