Competition · AMC preparation · step 4 of 4

AMC 12 2021 Fall A: 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 A #1 grade 6+ arithmetic
What is the value of ((2112-2021)²)/169?
AMC 12 2021 Fall A #2 grade 3+ geometry-2d
Menkara has a 4 × 6 index card. If she shortens the length of one side of this card by 1 inch, the card would have area…
AMC 12 2021 Fall A #3 grade 6+ rate-ratio
Mr. Lopez has a choice of two routes to get to work. Route A is 6 miles long, and his average speed along this route is…
AMC 12 2021 Fall A #4 grade 6+ number-theory
The six-digit number 2 0 2 1 0 A is prime for only one digit A. What is A?
AMC 12 2021 Fall A #5 grade 5+ rate-ratio
Elmer the emu takes 44 equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can…
AMC 12 2021 Fall A #6 grade 8+ geometry-2d
As shown in the figure below, point E lies on the opposite half-plane determined by line CD from point A so that ∠ CDE =…
AMC 12 2021 Fall A #7 grade 7+ probability
A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is tea…
AMC 12 2021 Fall A #8 grade 6+ number-theory
Let M be the least common multiple of all the integers 10 through 30, inclusive. Let N be the least common multiple of M…
AMC 12 2021 Fall A #9 grade 11+ geometry-3d, algebra
A right rectangular prism whose surface area and volume are numerically equal has edge lengths log₂x, log₃x, and log₄x.…
AMC 12 2021 Fall A #10 grade 8+ number-theory
The base-nine representation of the number N is 27,006,000,052_nine. What is the remainder when N is divided by 5?
AMC 12 2021 Fall A #11 grade 10+ geometry-2d
Consider two concentric circles of radius 17 and 19. The larger circle has a chord, half of which lies inside the smalle…
AMC 12 2021 Fall A #12 grade 12+ counting
What is the number of terms with rational coefficients among the 1001 terms in the expansion of (x³√2+y√3)¹⁰⁰⁰?
AMC 12 2021 Fall A #13 grade 10+ geometry-2d
The angle bisector of the acute angle formed at the origin by the graphs of the lines y = x and y=3x has equation y=kx.…
AMC 12 2021 Fall A #14 grade 10+ geometry-2d
In the figure, equilateral hexagon ABCDEF has three nonadjacent acute interior angles that each measure 30°. The enclose…
AMC 12 2021 Fall A #15 grade 11+ algebra
Recall that the conjugate of the complex number w = a + bi, where a and b are real numbers and i = √(-1), is the complex…
AMC 12 2021 Fall A #16 grade 7+ counting
An organization has 30 employees, 20 of whom have a brand A computer while the other 10 have a brand B computer. For sec…
AMC 12 2021 Fall A #17 grade 9+ algebra
For how many ordered pairs (b,c) of positive integers does neither x²+bx+c=0 nor x²+cx+b=0 have two distinct real soluti…
AMC 12 2021 Fall A #18 grade 11+ probability
Each of the 20 balls is tossed independently and at random into one of the 5 bins. Let p be the probability that some bi…
AMC 12 2021 Fall A #19 grade 11+ algebra
Let x be the least real number greater than 1 such that sin(x)= sin(x²), where the arguments are in degrees. What is x r…
AMC 12 2021 Fall A #20 grade 9+ number-theory
For each positive integer n, let f₁(n) be twice the number of positive integer divisors of n, and for j ≥ 2, let f_j(n)…
AMC 12 2021 Fall A #21 grade 8+ geometry-2d
Let ABCD be an isosceles trapezoid with BC∥ AD and AB=CD. Points X and Y lie on diagonal AC with X between A and Y, as s…
AMC 12 2021 Fall A #22 grade 11+ counting
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in a 3-by-3 array of boxes, then Carl pla…
AMC 12 2021 Fall A #23 grade 11+ algebra
A quadratic polynomial with real coefficients and leading coefficient 1 is called disrespectful if the equation p(p(x))=…
AMC 12 2021 Fall A #24 grade 11+ geometry-2d
Convex quadrilateral ABCD has AB = 18, ∠A = 60°, and AB ∥ CD. In some order, the lengths of the four sides form an arith…
AMC 12 2021 Fall A #25 grade 11+ counting, number-theory
Let m≥ 5 be an odd integer, and let D(m) denote the number of quadruples (a₁, a₂, a₃, a₄) of distinct integers with 1≤ a…

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