Competition · AMC preparation · step 4 of 4

AMC 10 2022B: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 10 2022B #1 grade 6+ arithmetic
Define xdiamond y to be |x-y| for all real numbers x and y. What is the value of (1diamond(2diamond3))-((1diamond2)diamo…
AMC 10 2022B #2 grade 8+ geometry-2d
In rhombus ABCD, point P lies on segment AD so that BP ⊥ AD, AP = 3, and PD = 2. What is the area of ABCD? (Note: The fi…
AMC 10 2022B #3 grade 4+ arithmetic
How many three-digit positive integers have an odd number of even digits?
AMC 10 2022B #4 grade 4+ rate-ratio
A donkey suffers an attack of hiccups and the first hiccup happens at 4:00 one afternoon. Suppose that the donkey hiccup…
AMC 10 2022B #5 grade 8+ arithmetic
What is the value of (1+1/3)(1+1/5)(1+1/7)/(√(1-1/3²)(1-1/5²)(1-1/7²))?
AMC 10 2022B #6 grade 5+ number-theory
How many of the first ten numbers of the sequence 121, 11211, 1112111, … are prime numbers?
AMC 10 2022B #7 grade 7+ algebra
For how many values of the constant k will the polynomial x²+kx+36 have two distinct integer roots?
AMC 10 2022B #8 grade 5+ arithmetic
Consider the following 100 sets of 10 elements each: {1,2,3,…,10}, {11,12,13,…,20}, {21,22,23,…,30}, ⋮ {991,992,993,…,10…
AMC 10 2022B #9 grade 7+ arithmetic
The 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…
AMC 10 2022B #10 grade 6+ arithmetic
Camila writes down five positive integers. The unique mode of these integers is 2 greater than their median, and the med…
AMC 10 2022B #11 grade 6+ arithmetic
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices belo…
AMC 10 2022B #12 grade 7+ probability
A 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 th…
AMC 10 2022B #13 grade 8+ number-theory
The positive difference between a pair of primes is equal to 2, and the positive difference between the cubes of the two…
AMC 10 2022B #14 grade 6+ arithmetic
Suppose that S is a subset of { 1, 2, 3, … , 25 } such that the sum of any two (not necessarily distinct) elements of S…
AMC 10 2022B #15 grade 8+ arithmetic
Let S_n be the sum of the first n terms of an arithmetic sequence that has a common difference of 2. The quotient (S_3n)…
AMC 10 2022B #16 grade 8+ geometry-2d
The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5. Three vertices of the squ…
AMC 10 2022B #17 grade 8+ number-theory
One of the following numbers is not divisible by any prime number less than 10. Which is it?
AMC 10 2022B #18 grade 8+ algebra
Consider systems of three linear equations with unknowns x, y, and z, a₁ x + b₁ y + c₁ z = 0 a₂ x + b₂ y + c₂ z = 0 a₃ x…
AMC 10 2022B #19 grade 3+ geometry-2d
Each square in a 5 × 5 grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighbori…
AMC 10 2022B #20 grade 8+ geometry-2d
Let 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 perpen…
AMC 10 2022B #21 grade 8+ number-theory
Let P(x) be a polynomial with rational coefficients such that when P(x) is divided by the polynomial x² + x + 1, the rem…
AMC 10 2022B #22 grade 8+ arithmetic
Let S be the set of circles in the coordinate plane that are tangent to each of the three circles with equations x²+y²=4…
AMC 10 2022B #23 grade 7+ probability
Ant Amelia starts on the number line at 0 and crawls in the following manner. For n=1,2,3, Amelia chooses a time duratio…
AMC 10 2022B #24 grade 8+ algebra
Consider 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 sa…
AMC 10 2022B #25 grade 6+ arithmetic
Let x₀,x₁,x₂,… be a sequence of numbers, where each x_k is either 0 or 1. For each positive integer n, define S_n = sum_…

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