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2022 AMC 10B problems

All 25 problems from the 2022 AMC 10B, 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 1Define x~♦~ y to be |x-y| for all real numbers x and y. What is the value of (1~♦~(2~♦~3))-((1~♦~2)~♦~3)?Algebra
  2. 2Problem 2In rhombus ABCD, point P lies on segment AD so that BP ⊥ AD, AP = 3, and PD = 2. What is the area of ABCD?Geometry
  3. 3Problem 3How many three-digit positive integers have an odd number of even digits?Number Theory
  4. 4Problem 4A donkey suffers an attack of hiccups and the first hiccup happens at 4:00 one afternoon. Suppose that the donkey hiccups regularly every 5 seconds…Algebra
  5. 5Problem 5What is the value of (1+1/3)(1+frac 15)(1+frac 17)/√((1-1/3^2)(1-1/5^2)(1-1/7^2))?Algebra
  6. 6Problem 6How many of the first ten numbers of the sequence 121, 11211, 1112111, … are prime numbers?Algebra
  7. 7Problem 7For how many values of the constant k will the polynomial x^2+kx+36 have two distinct integer roots?Algebra
  8. 8Problem 8Consider the following 100 sets of 10 elements each: {1,2,3,…,10}, {11,12,13,…,20}, {21,22,23,…,30}, vdots {991,992,993,…,1000}. How many of these…Number Theory
  9. 9Problem 9The sum 1/2!+2/3!+3/4!+…+2021/2022! can be expressed as a-1/b!, where a and b are positive integers. What is a+b?Algebra
  10. 10Problem 10Camila writes down five positive integers. The unique mode of these integers is 2 greater than their median, and the median is 2 greater than their…Algebra
  11. 11Problem 11All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to…
  12. 12Problem 12A pair of fair 6-sided dice is rolled n times. What is the least value of n such that the probability that the sum of the numbers face up on a roll…Counting & Probability
  13. 13Problem 13The positive difference between a pair of primes is equal to 2, and the positive difference between the cubes of the two primes is 31106. What is the…Number Theory
  14. 14Problem 14Suppose that S is a subset of { 1, 2, 3, …, 25 } such that the sum of any two (not necessarily distinct) elements of S is never an element of S. What…Counting & Probability
  15. 15Problem 15Let S_n be the sum of the first n terms of an arithmetic sequence that has a common difference of 2. The quotient dfrac S_3nS_n does not depend on n…Algebra
  16. 16Problem 16The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5. Three vertices of the square lie on three different…Geometry
  17. 17Problem 17One of the following numbers is not divisible by any prime number less than 10. Which is it?Number Theory
  18. 18Problem 18Consider systems of three linear equations with unknowns x, y, and z, a_1 x + b_1 y + c_1 z = 0 a_2 x + b_2 y + c_2 z = 0 a_3 x + b_3 y + c_3 z = 0…Algebra
  19. 19Problem 19Each square in a 5 × 5 grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a…
  20. 20Problem 20Let ABCD be a rhombus with ∠ ADC = 46°. Let E be the midpoint of CD, and let F be the point on BE such that AF is perpendicular to BE. What is the…Geometry
  21. 21Problem 21Let P(x) be a polynomial with rational coefficients such that when P(x) is divided by the polynomial x^2 + x + 1, the remainder is x+2, and when P(x)…Algebra
  22. 22Problem 22Let S be the set of circles in the coordinate plane that are tangent to each of the three circles with equations x^2+y^2=4, x^2+y^2=64, and…Geometry
  23. 23Problem 23Ant Amelia starts on the number line at 0 and crawls in the following manner. For n=1, 2, 3; Amelia chooses a time duration t_n and an increment x_n…Counting & Probability
  24. 24Problem 24Consider functions f that satisfy |f(x)-f(y)|≤ 1/2|x-y| for all real numbers x and y. Of all such functions that also satisfy the equation f(300) =…Algebra
  25. 25Problem 25Let x_0, x_1, x_2, dotsc be a sequence of numbers, where each x_k is either 0 or 1. For each positive integer n, define S_n = sum _k=0^n-1 x_k 2^k…Number Theory

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