Skip to main content

2023 AMC 10A problems

All 25 problems from the 2023 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 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 1/3 of a large pizza together with 31/2 cups of orange slices is the same as the weight of 3/4 of a large pizza together with 1/2 cup…Algebra
  3. 3Problem 3How many positive perfect squares less than 2023 are divisible by 5?Number Theory
  4. 4Problem 4A quadrilateral has all integer side lengths, a perimeter of 26, and one side of length 4. What is the greatest possible length of one side of this…Algebra
  5. 5Problem 5How many digits are in the base-ten representation of 8^5 · 5^10 · 15^5?Number Theory
  6. 6Problem 6An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the…Geometry
  7. 7Problem 7Janet 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 running…Counting & Probability
  8. 8Problem 8Barb the baker creates a new temperature system for baking bread, Breadus, which is linearly based on Fahrenheit. Bread rises at 110 F^°, which is 0…Algebra
  9. 9Problem 9A 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
  10. 10Problem 10Maureen 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
  11. 11Problem 11A 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
  12. 12Problem 12How many three-digit positive integers N satisfy the following properties? • The number N is divisible by 7. • The number formed by reversing the…Number Theory
  13. 13Problem 13Abdul and Chiang are standing 48 feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed…Geometry
  14. 14Problem 14A number is chosen at random from among the first 100 positive integers, and a positive integer divisor of that number is then chosen at random. What…Number Theory
  15. 15Problem 15An even number of circles are nested, starting with a radius of 1 and increasing by 1 each time, all sharing a common point. The region between every…Algebra
  16. 16Problem 16In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as…Algebra
  17. 17Problem 17Let ABCD be a rectangle with AB = 30 and BC = 28. Points P and Q lie on BC and CD respectively so that all sides of △ ABP, △ PCQ, and △ QDA have…Geometry
  18. 18Problem 18A rhombic dodecahedron is a solid with 12 congruent rhombus faces. At every vertex, 3 or 4 edges meet, depending on the vertex. How many vertices…Geometry
  19. 19Problem 19The line segment formed by A(1, 2) and B(3, 3) is rotated to the line segment formed by A'(3, 1) and B'(4, 3) about the point P(r, s). What is |r -…Geometry
  20. 20Problem 20Each square in a 3 × 3 grid of squares is colored red, white, blue, or green so that every 2 × 2 square contains one square of each color. One such…Counting & Probability
  21. 21Problem 21There is a unique polynomial P(x) of least degree with leading coefficient 1 satisfying all of the following: 1 is a root of P(x) - 1, 2 is a root of…Algebra
  22. 22Problem 22Circle C_1 and C_2 each have radius 1, and the distance between their centers is 1/2. Circle C_3 is the largest circle internally tangent to both C_1…Geometry
  23. 23Problem 23Positive integer divisors a and b of N are called complementary if ab = N. Given that N has a pair of complementary divisors that differ by 20 and a…Number Theory
  24. 24Problem 24Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame…Geometry
  25. 25Problem 25If 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…Counting & Probability

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