Competition · AMC preparation · step 4 of 4

AMC 12 2020A: 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 2020A #1 grade 6+ arithmetic
Carlos took 70% of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
AMC 12 2020A #2 grade 8+ geometry-2d
The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of…
AMC 12 2020A #3 grade 6+ rate-ratio
A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is pai…
AMC 12 2020A #4 grade 4+ number-theory
How many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divi…
AMC 12 2020A #5 grade 6+ arithmetic
The 25 integers from -10 to 14, inclusive, can be arranged to form a 5-by-5 square in which the sum of the numbers in ea…
AMC 12 2020A #6 grade 5+ geometry-2d
In the plane figure shown below, 3 of the unit squares have been shaded. What is the least number of additional unit squ…
AMC 12 2020A #7 grade 8+ geometry-3d
Seven cubes, whose volumes are 1, 8, 27, 64, 125, 216, and 343 cubic units, are stacked vertically to form a tower in wh…
AMC 12 2020A #8 grade 6+ arithmetic
What is the median of the following list of 4040 numbers? 1, 2, 3, …, 2020, 1², 2², 3², …, 2020²
AMC 12 2020A #9 grade 11+ algebra
How many solutions does the equation tan(2x)=cos(x/2) have on the interval [0,2π]?
AMC 12 2020A #10 grade 11+ algebra
There is a unique positive integer n such thatlog₂(log₁₆n) = log₄(log₄n).What is the sum of the digits of n?
AMC 12 2020A #11 grade 7+ probability
A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes…
AMC 12 2020A #12 grade 10+ geometry-2d
Line l in the coordinate plane has equation 3x-5y+40=0. This line is rotated 45° counterclockwise about the point (20,20…
AMC 12 2020A #13 grade 11+ algebra
There are integers a, b, and c, each greater than 1, such that ^a√(N^b√(N^c√N)) = ³⁶√N²⁵ for all N ≠ 1. What is b?
AMC 12 2020A #14 grade 10+ geometry-2d
Regular octagon ABCDEFGH has area n. Let m be the area of quadrilateral ACEG. What is m/n?
AMC 12 2020A #15 grade 11+ algebra
In the complex plane, let A be the set of solutions to z³-8=0 and let B be the set of solutions to z³-8z²-8z+64=0. What…
AMC 12 2020A #16 grade 8+ geometry-2d
A point is chosen at random within the square in the coordinate plane whose vertices are (0, 0), (2020, 0), (2020, 2020)…
AMC 12 2020A #17 grade 11+ geometry-2d
The vertices of a quadrilateral lie on the graph of y=lnx, and the x-coordinates of these vertices are consecutive posit…
AMC 12 2020A #18 grade 8+ geometry-2d
Quadrilateral ABCD satisfies ∠ ABC = ∠ ACD = 90°, AC=20, and CD=30. Diagonals AC and BD intersect at point E, and AE=5.…
AMC 12 2020A #19 grade 8+ number-theory
There exists a unique strictly increasing sequence of nonnegative integers a₁ < a₂ < … < a_k such that(2²⁸⁹+1)/(2¹⁷+1) =…
AMC 12 2020A #20 grade 8+ geometry-2d
Let T be the triangle in the coordinate plane with vertices (0,0), (4,0), and (0,3). Consider the following five isometr…
AMC 12 2020A #21 grade 8+ number-theory
How many positive integers n are there such that n is a multiple of 5, and the least common multiple of 5! and n equals…
AMC 12 2020A #22 grade 11+ algebra
Let (a_n) and (b_n) be the sequences of real numbers such that (2 + i)ⁿ = a_n + b_ni for all integers n≥ 0, where i = √(…
AMC 12 2020A #23 grade 7+ probability
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly em…
AMC 12 2020A #24 grade 8+ geometry-2d
Suppose that △ABC is an equilateral triangle of side length s, with the property that there is a unique point P inside t…
AMC 12 2020A #25 grade 9+ algebra
The number a=p/q, where p and q are relatively prime positive integers, has the property that the sum of all real number…

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