Competition · AMC preparation · step 4 of 4

AMC 12 2021 Fall B: 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 2021 Fall B #1 grade 4+ arithmetic
What is the value of 1234 + 2341 + 3412 + 4123
AMC 12 2021 Fall B #2 grade 6+ geometry-2d
What is the area of the shaded figure shown below?
AMC 12 2021 Fall B #3 grade 7+ algebra
At noon on a certain day, Minneapolis is N degrees warmer than St. Louis. At 4:00 the temperature in Minneapolis has fal…
AMC 12 2021 Fall B #4 grade 8+ algebra
Let n=8²⁰²². Which of the following is equal to n/4?
AMC 12 2021 Fall B #5 grade 7+ number-theory, counting
Call a fraction a/b, not necessarily in the simplest form, special if a and b are positive integers whose sum is 15. How…
AMC 12 2021 Fall B #6 grade 9+ number-theory
The greatest prime number that is a divisor of 16,384 is 2 because 16,384 = 2¹⁴. What is the sum of the digits of the gr…
AMC 12 2021 Fall B #7 grade 9+ algebra
Which of the following conditions is sufficient to guarantee that integers x, y, and z satisfy the equation x(x-y)+y(y-z…
AMC 12 2021 Fall B #8 grade 11+ geometry-2d
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the bas…
AMC 12 2021 Fall B #9 grade 8+ geometry-2d
Triangle ABC is equilateral with side length 6. Suppose that O is the center of the inscribed circle of this triangle. W…
AMC 12 2021 Fall B #10 grade 11+ geometry-2d
What is the sum of all possible values of t between 0 and 360 such that the triangle in the coordinate plane whose verti…
AMC 12 2021 Fall B #11 grade 7+ probability
Una rolls 6 standard 6-sided dice simultaneously and calculates the product of the 6 numbers obtained. What is the proba…
AMC 12 2021 Fall B #12 grade 6+ number-theory
For n a positive integer, let f(n) be the quotient obtained when the sum of all positive divisors of n is divided by n.…
AMC 12 2021 Fall B #13 grade 11+ algebra
Let c = 2π/11. What is the value of (sin 3c · sin 6c · sin 9c · sin 12c · sin 15c)/(sin c · sin 2c · sin 3c · sin 4c · s…
AMC 12 2021 Fall B #14 grade 11+ algebra
Suppose that P(z), Q(z), and R(z) are polynomials with real coefficients, having degrees 2, 3, and 6, respectively, and…
AMC 12 2021 Fall B #15 grade 10+ geometry-2d
Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rot…
AMC 12 2021 Fall B #16 grade 6+ number-theory
Suppose a, b, c are positive integers such that a+b+c=23 and gcd(a,b)+gcd(b,c)+gcd(c,a)=9. What is the sum of all possib…
AMC 12 2021 Fall B #17 grade 8+ probability
A bug starts at a vertex of a grid made of equilateral triangles of side length 1. At each step the bug moves in one of…
AMC 12 2021 Fall B #18 grade 9+ algebra
Set u₀ = 1/4, and for k ≥ 0 let u_(k+1) be determined by the recurrence u_(k+1) = 2u_k - 2u_k². This sequence tends to a…
AMC 12 2021 Fall B #19 grade 10+ counting, geometry-2d
Regular polygons with 5,6,7, and 8 sides are inscribed in the same circle. No two of the polygons share a vertex, and no…
AMC 12 2021 Fall B #20 grade 8+ counting, geometry-3d
A cube is constructed from 4 white unit cubes and 4 blue unit cubes. How many different ways are there to construct the…
AMC 12 2021 Fall B #21 grade 11+ algebra
For real numbers x, let P(x)=1+cos(x)+isin(x)-cos(2x)-isin(2x)+cos(3x)+isin(3x) where i = √(-1). For how many values of…
AMC 12 2021 Fall B #22 grade 10+ geometry-2d
Right triangle ABC has side lengths BC=6, AC=8, and AB=10. A circle centered at O is tangent to line BC at B and passes…
AMC 12 2021 Fall B #23 grade 12+ probability, counting
What is the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen…
AMC 12 2021 Fall B #24 grade 11+ geometry-2d
Triangle ABC has side lengths AB = 11, BC=24, and CA = 20. The bisector of ∠BAC intersects BC in point D, and intersects…
AMC 12 2021 Fall B #25 grade 6+ number-theory
For n a positive integer, let R(n) be the sum of the remainders when n is divided by 2, 3, 4, 5, 6, 7, 8, 9, and 10. For…

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