Competition · AMC preparation · step 4 of 4

AMC 12 2019A: 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 2019A #1 grade 7+ geometry-2d
The area of a pizza with radius 4 inches is N percent larger than the area of a pizza with radius 3 inches. What is the…
AMC 12 2019A #2 grade 7+ rate-ratio
Suppose a is 150% of b. What percent of a is 3b?
AMC 12 2019A #3 grade 3+ arithmetic
A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is…
AMC 12 2019A #4 grade 6+ number-theory
What is the greatest number of consecutive integers whose sum is 45?
AMC 12 2019A #5 grade 6+ geometry-2d
Two lines with slopes 1/2 and 2 intersect at (2,2). What is the area of the triangle enclosed by these two lines and the…
AMC 12 2019A #6 grade 8+ geometry-2d
The figure below shows line ell with a regular, infinite, recurring pattern of squares and line segments. How many of th…
AMC 12 2019A #7 grade 6+ arithmetic
Melanie computes the mean μ, the median M, and the modes of the 365 values that are the dates in the months of 2019. Thu…
AMC 12 2019A #8 grade 8+ counting
For a set of four distinct lines in a plane, there are exactly N distinct points that lie on two or more of the lines. W…
AMC 12 2019A #9 grade 8+ number-theory
A sequence of numbers is defined recursively by a₁ = 1, a₂ = 3/7, and a_n=(a_(n-2) · a_(n-1))/(2a_(n-2) - a_(n-1))for al…
AMC 12 2019A #10 grade 8+ geometry-2d
The figure below shows 13 circles of radius 1 within a larger circle. All the intersections occur at points of tangency.…
AMC 12 2019A #11 grade 8+ number-theory
For some positive integer k, the repeating base-k representation of the (base-ten) fraction 7/51 is 0.23_k = 0.232323...…
AMC 12 2019A #12 grade 11+ algebra
Positive real numbers x ≠ 1 and y ≠ 1 satisfy log₂x = log_y16 and xy = 64. What is (log₂x/y)²?
AMC 12 2019A #13 grade 7+ counting
How many ways are there to paint each of the integers 2, 3, … , 9 either red, green, or blue so that each number has a d…
AMC 12 2019A #14 grade 11+ algebra
For a certain complex number c, the polynomial P(x) = (x² - 2x + 2)(x² - cx + 4)(x² - 4x + 8)has exactly 4 distinct root…
AMC 12 2019A #15 grade 11+ algebra
Positive real numbers a and b have the property that √loga + √logb + log √a + log √b = 100 and all four terms on the lef…
AMC 12 2019A #16 grade 7+ probability
The numbers 1,2,…,9 are randomly placed into the 9 squares of a 3 × 3 grid. Each square gets one number, and each of the…
AMC 12 2019A #17 grade 11+ algebra
Let s_k denote the sum of the kth powers of the roots of the polynomial x³-5x²+8x-13. In particular, s₀=3, s₁=5, and s₂=…
AMC 12 2019A #18 grade 8+ geometry-3d
A sphere with center O has radius 6. A triangle with sides of length 15, 15, and 24 is situated in space so that each of…
AMC 12 2019A #19 grade 11+ geometry-2d
In △ ABC with integer side lengths, cos A = 11/16, cos B = 7/8, and cos C = -1/4. What is the least possible perimeter f…
AMC 12 2019A #20 grade 7+ probability
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads,…
AMC 12 2019A #21 grade 11+ algebra
Let z=(1+i)/√2.What is (z^1²+z^2²+z^3²+…+z^12²) · (1/(z^1²)+1/(z^2²)+1/(z^3²)+…+1/(z^12²))?
AMC 12 2019A #22 grade 11+ geometry-2d
Circles ω and γ, both centered at O, have radii 20 and 17, respectively. Equilateral triangle ABC, whose interior lies i…
AMC 12 2019A #23 grade 11+ algebra
Define binary operations ♢ and ♡ by a ♢ b = a^(log₇(b)) and a ♡ b = a^(1/(log₇(b)))for all real numbers a and b for whic…
AMC 12 2019A #24 grade 8+ number-theory
For how many integers n between 1 and 50, inclusive, is ((n²-1)!)/((n!)ⁿ) an integer? (Recall that 0! = 1.)
AMC 12 2019A #25 grade 10+ geometry-2d
Let △ A₀B₀C₀ be a triangle whose angle measures are exactly 59.999°, 60°, and 60.001°. For each positive integer n, defi…

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