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2024 AMC 12A problems

All 25 problems from the 2024 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 1What is the value of 9901cdot 101-99cdot 10101?Algebra
  2. 2Problem 2A model used to estimate the time it will take to hike to the top of a mountain on a trail is of the form T=aL+bG, where a and b are constants, T is…Algebra
  3. 3Problem 3The number 2024 is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write…Algebra
  4. 4Problem 4What is the least value of n such that n! is a multiple of 2024?Number Theory
  5. 5Problem 5A data set containing 20 numbers, some of which are 6, has mean 45. When all the 6s are removed, the data set has mean 66. How many 6s were in the…Algebra
  6. 6Problem 6The product of three integers is 60. What is the least possible positive sum of the three integers?Algebra
  7. 7Problem 7In △ ABC, ∠ ABC=90° and BA=BC=sqrt 2. Points P_1, P_2, …, P_2024 lie on hypotenuse AC so that AP_1=P_1P_2 =P_2P_3 =… =P_2023P_2024 =P_2024C. What is…Geometry
  8. 8Problem 8How many angles θ with 0≤θle 2π satisfy log (sin (3θ))+log (cos (2θ))=0?Algebra
  9. 9Problem 9Let M be the greatest integer such that both M+1213 and M+3773 are perfect squares. What is the units digit of M?Algebra
  10. 10Problem 10Let alpha be the radian measure of the smallest angle in a 3-4-5 right triangle. Let beta be the radian measure of the smallest angle in a 7-24-25…Geometry
  11. 11Problem 11There are exactly K positive integers b with 5≤ ble 2024 such that the base-b integer 2024_b is divisible by 16 (where 16 is in base ten). What is…Number Theory
  12. 12Problem 12The first three terms of a geometric sequence are the integers a, 720, and b, where alt 720< b. What is the sum of the digits of the least possible…Algebra
  13. 13Problem 13The graph of y=e^x+1+e^-x-2 has an axis of symmetry. What is the reflection of the point (-1,tfrac 12) over this axis?Algebra
  14. 14Problem 14The numbers, in order, of each row and the numbers, in order, of each column of a 5times 5 array of integers form an arithmetic progression of length…Algebra
  15. 15Problem 15The roots of x^3+2x^2-x+3 are p, q, and r. What is the value of (p^2+4)(q^2+4)(r^2+4)?Algebra
  16. 16Problem 16A set of 12 tokens — 3 red, 2 white, 1 blue, and 6 black — is to be distributed at random to 3 game players, 4 tokens per player. The probability…Counting & Probability
  17. 17Problem 17Integers a, b, and c satisfy ab+c=100, bc+a=87, and ca+b=60. What is ab+bc+ca?Algebra
  18. 18Problem 18On top of a rectangular card with sides of length 1 and 2+sqrt 3, an identical card is placed so that two of their diagonals line up, as shown (AC…Geometry
  19. 19Problem 19Cyclic quadrilateral ABCD has lengths BC=CD=3 and DA=5 with ∠ CDA=120°. What is the length of the shorter diagonal of ABCD?Geometry
  20. 20Problem 20Points P and Q are chosen uniformly and independently at random on sides AB and AC, respectively, of equilateral triangle △ ABC. Which of the…Geometry
  21. 21Problem 21Suppose that a_1=2 and the sequence (a_n) satisfies the recurrence relation a_n-1/n-1=frac a_n-1+1(n-1)+1 for all nge 2. What is the greatest integer…Algebra
  22. 22Problem 22The figure below shows a dotted grid 8 cells wide and 3 cells tall consisting of 1''times 1'' squares. Carl places 1-inch toothpicks along some of…Counting & Probability
  23. 23Problem 23What is the value of tan ^2π/16·tan ^23π/16 +tan ^2π/16·tan ^25π/16 +tan ^23π/16·tan ^27π/16 +tan ^25π/16·tan ^27π/16?Algebra
  24. 24Problem 24A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces…Geometry
  25. 25Problem 25A graph is symmetric about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers (a,b,c,d), where…Algebra

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