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2019 AMC 10A problems

All 25 problems from the 2019 AMC 10A, 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 1What is the value of 2^(0^(1^9))+((2^0)^1)^9?Algebra
  2. 2Problem 2What is the hundreds digit of (20!-15!)?Number Theory
  3. 3Problem 3Ana and Bonita were born on the same date in different years, n years apart. Last year Ana was 5 times as old as Bonita. This year Ana’s age is the…Algebra
  4. 4Problem 4A 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…Algebra
  5. 5Problem 5What is the greatest number of consecutive integers whose sum is 45?Algebra
  6. 6Problem 6For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four…Geometry
  7. 7Problem 7Two 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
  8. 8Problem 8The 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
  9. 9Problem 9What is the greatest three-digit positive integer n for which the sum of the first n positive integers is not a divisor of the product of the first n…Number Theory
  10. 10Problem 10A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one corner to the opposite corner…Number Theory
  11. 11Problem 11How many positive integer divisors of 201^9 are perfect squares or perfect cubes (or both)?Number Theory
  12. 12Problem 12Melanie 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…Algebra
  13. 13Problem 13Let △ ABC be an isosceles triangle with BC = AC and ∠ ACB = 40^°. Construct the circle with diameter BC, and let D and E be the other intersection…Geometry
  14. 14Problem 14For 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
  15. 15Problem 15A 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
  16. 16Problem 16The 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
  17. 17Problem 17A child builds towers using identically shaped cubes of different colors. How many different towers with a height 8 cubes can the child build with 2…Counting & Probability
  18. 18Problem 18For some positive integer k, the repeating base-k representation of the (base-ten) fraction 7/51 is 0.23_k = 0.232323..._k. What is k?Number Theory
  19. 19Problem 19What is the least possible value of (x+1)(x+2)(x+3)(x+4) +2019, where x is a real number?Algebra
  20. 20Problem 20The 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…Number Theory
  21. 21Problem 21A 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
  22. 22Problem 22Real 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
  23. 23Problem 23Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will…Algebra
  24. 24Problem 24Let p, q, and r be the distinct roots of the polynomial x^3 - 22x^2 + 80x - 67. There exist real numbers A, B, and C such that 1/s^3-22s^2+80s-67…Algebra
  25. 25Problem 25For how many integers n between 1 and 50, inclusive, is (n^2-1)!/(n!)^n an integer? (Recall that 0! = 1.)Number Theory

Practise the same ideas across every year on the topic pages, or browse the full AMC 10 archive.