Skip to main content

2023 AMC 12A problems

All 25 problems from the 2023 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 1Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at 18 miles per hour. Leaving at the same time, Beth…Algebra
  2. 2Problem 2The weight of tfrac 13 of a large pizza together with 3tfrac 12 cups of orange slices is the same as the weight of tfrac 34 of a large pizza together…Algebra
  3. 3Problem 3How many positive perfect squares less than 2023 are divisible by 5?Number Theory
  4. 4Problem 4How many digits are in the base-ten representation of 8^5· 5^10· 15^5?Number Theory
  5. 5Problem 5Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her…Counting & Probability
  6. 6Problem 6Points A and B lie on the graph of y=log _2 x. The midpoint of AB is (6,2). What is the positive difference between the x-coordinates of A and B?Algebra
  7. 7Problem 7A digital display shows the current date as an 8-digit integer consisting of a 4-digit year, followed by a 2-digit month, followed by a 2-digit date…Number Theory
  8. 8Problem 8Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an 11 on the next quiz, her mean will increase by 1. If she…Algebra
  9. 9Problem 9A square of area 2 is inscribed in a square of area 3, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the…Geometry
  10. 10Problem 10Positive real numbers x and y satisfy y^3=x^2 and (y-x)^2=4y^2. What is x+y?Algebra
  11. 11Problem 11What is the degree measure of the acute angle formed by lines with slopes 2 and tfrac 13?Geometry
  12. 12Problem 12What is the value of 2^3-1^3+4^3-3^3+6^3-5^3 +…+18^3-17^3?Algebra
  13. 13Problem 13In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as…Number Theory
  14. 14Problem 14How many complex numbers satisfy the equation z^5=z, where z is the conjugate of the complex number z?Algebra
  15. 15Problem 15Usain is walking for exercise by zigzagging across a 100-meter by 30-meter rectangular field, beginning at point A and ending on the segment BC. He…Geometry
  16. 16Problem 16Consider the set of complex numbers z satisfying |1+z+z^2|=4. The maximum value of the imaginary part of z can be written in the form √(m)/n, where m…Algebra
  17. 17Problem 17Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps…Algebra
  18. 18Problem 18Circle C_1 and C_2 each have radius 1, and the distance between their centers is tfrac 12. Circle C_3 is the largest circle internally tangent to…Geometry
  19. 19Problem 19What is the product of all the solutions to the equation log _7x2023·log _289x2023 =log _2023x2023?Algebra
  20. 20Problem 20Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below. ccccccccc 1 1 1 1 3 1 1 5 5 1 1 7 11 7 1 Each row after the first row is…Algebra
  21. 21Problem 21If A and B are vertices of a polyhedron, define the distance d(A,B) to be the minimum number of edges of the polyhedron one must traverse in order to…Counting & Probability
  22. 22Problem 22Let f be the unique function defined on the positive integers such that sum _dmid n d· f(n/d)=1 for all positive integers n, where the sum is taken…Algebra
  23. 23Problem 23How many ordered pairs of positive real numbers (a,b) satisfy the equation (1+2a)(2+2b)(2a+b) =32ab?Algebra
  24. 24Problem 24Let K be the number of sequences A_1, A_2, …, A_n such that n is a positive integer less than or equal to 10, each A_i is a subset of {1,2,3,…,10}…Counting & Probability
  25. 25Problem 25There is a unique sequence of integers a_1, a_2, … a_2023 such that tan 2023x =tiny dfrac a_1tan x+a_3tan ^3 x+a_5tan ^5 x+…+a_2023tan ^2023x1+a_2tan…Algebra

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