Skip to main content

2019 AMC 12A problems

All 25 problems from the 2019 AMC 12A, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1The 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 integer closest to N?Algebra
  2. 2Problem 2Suppose a is 150% of b. What percent of a is 3b?Algebra
  3. 3Problem 3A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is the minimum number of balls…Counting & Probability
  4. 4Problem 4What is the greatest number of consecutive integers whose sum is 45?Algebra
  5. 5Problem 5Two 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 line x + y = 10?Geometry
  6. 6Problem 6The figure below shows line ℓ with a regular, infinite, recurring pattern of squares and line segments. How many of the following four kinds of rigid…Geometry
  7. 7Problem 7Melanie computes the mean mu, the median M, and the modes of the 365 values that are the dates in the months of 2019. Thus her data consist of 12 1s…Algebra
  8. 8Problem 8For a set of four distinct lines in a plane, there are exactly N distinct points that lie on two or more of the lines. What is the sum of all…Counting & Probability
  9. 9Problem 9A sequence of numbers is defined recursively by a_1 = 1, a_2 = 3/7, and a_n = dfrac a_n-2 · a_n-12a_n-2 - a_n-1 for all n ≥ 3. Then a_2019 can be…Algebra
  10. 10Problem 10The figure below shows 13 circles of radius 1 within a larger circle. All the intersections occur at points of tangency. What is the area of the…Geometry
  11. 11Problem 11For some positive integer k, the repeating base-k representation of the (base-ten) fraction 7/51 is 0.23_k = 0.232323ldots _k. What is k?Number Theory
  12. 12Problem 12Positive real numbers x ≠ 1 and y ≠ 1 satisfy log _2 x = log _y 16 and xy = 64. What is (log _2 x/y)^2?Algebra
  13. 13Problem 13How many ways are there to paint each of the integers 2, 3, …, 9 either red, green, or blue so that each number has a different color from each of…Number Theory
  14. 14Problem 14For a certain complex number c, the polynomial P(x) = (x^2 - 2x + 2) · (x^2 - cx + 4) · (x^2 - 4x + 8) has exactly 4 distinct roots. What is |c|?Algebra
  15. 15Problem 15Positive real numbers a and b have the property that √(log a) + √(log b) + log √(a) + log √(b) = 100 and all four terms on the left are positive…Algebra
  16. 16Problem 16The 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 numbers is used once…Counting & Probability
  17. 17Problem 17Let s_k denote the sum of the kth powers of the roots of the polynomial x^3 - 5x^2 + 8x - 13. In particular, s_0 = 3, s_1 = 5, and s_2 = 9. Let a, b…Algebra
  18. 18Problem 18A 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 its sides is tangent to the…Geometry
  19. 19Problem 19In △ ABC with integer side lengths, cos A = 11/16, cos B = 7/8, cos C = -1/4. What is the least possible perimeter for △ ABC?Geometry
  20. 20Problem 20Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and…Counting & Probability
  21. 21Problem 21Let z = 1 + i/√(2). What is (z^1^2 + z^2^2 + z^3^2 + … + z^12^2) · scriptsize (dfrac 1z^1^2 + dfrac 1z^2^2 + dfrac 1z^3^2 + … + dfrac 1z^12^2)?Algebra
  22. 22Problem 22Circles ω and gamma, both centered at O, have radii 20 and 17, respectively. Equilateral triangle ABC, whose interior lies in the interior of ω but…Geometry
  23. 23Problem 23Define binary operations ♦ and heartsuit by a ♦ b = a^log _7(b) and a heartsuit b = a^1/log _7(b) for all real numbers a and b for which these…Algebra
  24. 24Problem 24For how many integers n between 1 and 50, inclusive, is (n^2 - 1)!/(n!)^n an integer? (Recall that 0! = 1.)Number Theory
  25. 25Problem 25Let △ A_0 B_0 C_0 be a triangle whose angle measures are exactly 59.999°, 60°, and 60.001°. For each positive integer n define A_n to be the foot of…Geometry

Practise the same ideas across every year on the topic pages, or browse the full AMC 12 archive.