Competition · AMC preparation · step 4 of 4

AMC 12 2020B: 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 2020B #1 grade 8+ arithmetic
What is the value in simplest form of the following expression?√1 + √(1+3) + √(1+3+5) + √(1+3+5+7)
AMC 12 2020B #2 grade 7+ algebra
What is the value of the following expression? (100²-7²)/(70²-11²) · (70-11)(70+11)/(100-7)(100+7)
AMC 12 2020B #3 grade 6+ rate-ratio
The ratio of w to x is 4:3, the ratio of y to z is 3:2, and the ratio of z to x is 1:6. What is the ratio of w to y?
AMC 12 2020B #4 grade 8+ geometry-2d
The acute angles of a right triangle are a° and b°, where a>b and both a and b are prime numbers. What is the least poss…
AMC 12 2020B #5 grade 8+ algebra
Teams A and B are playing in a basketball league where each game results in a win for one team and a loss for the other…
AMC 12 2020B #6 grade 6+ algebra
For all integers n ≥ 9, the value of ((n+2)!-(n+1)!)/(n!)is always which of the following?
AMC 12 2020B #7 grade 11+ geometry-2d
Two nonhorizontal, non vertical lines in the xy-coordinate plane intersect to form a 45° angle. One line has slope equal…
AMC 12 2020B #8 grade 6+ algebra
How many ordered pairs of integers (x, y) satisfy the equation x²⁰²⁰+y²=2y?
AMC 12 2020B #9 grade 8+ geometry-3d
A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral su…
AMC 12 2020B #10 grade 10+ geometry-2d
In unit square ABCD, the inscribed circle ω intersects CD at M, and AM intersects ω at a point P different from M. What…
AMC 12 2020B #11 grade 7+ geometry-2d
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the di…
AMC 12 2020B #12 grade 9+ geometry-2d
Let AB be a diameter in a circle of radius 5√2. Let CD be a chord in the circle that intersects AB at a point E such tha…
AMC 12 2020B #13 grade 11+ algebra
Which of the following is the value of √(log₂6+log₃6)?
AMC 12 2020B #14 grade 5+ logic
Bela and Jenn play the following game on the closed interval [0, n] of the real number line, where n is a fixed integer…
AMC 12 2020B #15 grade 4+ counting
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 pe…
AMC 12 2020B #16 grade 7+ probability
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the follo…
AMC 12 2020B #17 grade 11+ algebra
How many polynomials of the form x⁵ + ax⁴ + bx³ + cx² + dx + 2020, where a, b, c, and d are real numbers, have the prope…
AMC 12 2020B #18 grade 8+ geometry-2d
In square ABCD, points E and H lie on AB and DA, respectively, so that AE=AH. Points F and G lie on BC and CD, respectiv…
AMC 12 2020B #19 grade 8+ geometry-2d
Square ABCD in the coordinate plane has vertices at the points A(1,1), B(-1,1), C(-1,-1), and D(1,-1). Consider the foll…
AMC 12 2020B #20 grade 8+ probability
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at ra…
AMC 12 2020B #21 grade 8+ number-theory
How many positive integers n satisfy (n+1000)/70 = ⌊ √n ⌋?(Recall that ⌊ x⌋ is the greatest integer not exceeding x.)
AMC 12 2020B #22 grade 11+ algebra
What is the maximum value of ((2^t-3t)t)/(4^t) for real values of t?
AMC 12 2020B #23 grade 12+ algebra
How many integers n ≥ 2 are there such that whenever z₁, z₂, ..., z_n are complex numbers such that |z₁| = |z₂| = ... =…
AMC 12 2020B #24 grade 7+ counting
Let D(n) denote the number of ways of writing the positive integer n as a productn = f₁· f₂… f_k,where k≥1, the f_i are…
AMC 12 2020B #25 grade 11+ probability
For each real number a with 0 ≤ a ≤ 1, let numbers x and y be chosen independently at random from the intervals [0, a] a…

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