Competition · AMC preparation · step 4 of 4

AMC 12 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 12 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 12 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 12 2022B #3 grade 5+ number-theory
How many of the first ten numbers of the sequence 121, 11211, 1112111, … are prime numbers?
AMC 12 2022B #4 grade 7+ algebra
For how many values of the constant k will the polynomial x²+kx+36 have two distinct integer roots?
AMC 12 2022B #5 grade 8+ geometry-2d
The point (-1, -2) is rotated 270° counterclockwise about the point (3, 1). What are the coordinates of its new position…
AMC 12 2022B #6 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 12 2022B #7 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 12 2022B #8 grade 11+ algebra, geometry-2d
What is the graph of y⁴+1=x⁴+2y² in the coordinate plane?
AMC 12 2022B #9 grade 8+ algebra
The sequence a₀,a₁,a₂,… is a strictly increasing arithmetic sequence of positive integers such that 2^a₇=2²⁷ · a₇. What…
AMC 12 2022B #10 grade 8+ geometry-2d
Regular hexagon ABCDEF has side length 2. Let G be the midpoint of AB, and let H be the midpoint of DE. What is the peri…
AMC 12 2022B #11 grade 11+ algebra
Let f(n) = ( (-1+i√3)/2 )ⁿ + ( (-1-i√3)/2 )ⁿ, where i = √(-1). What is f(2022)?
AMC 12 2022B #12 grade 7+ probability, counting
Kayla rolls four fair 6-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than…
AMC 12 2022B #13 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 12 2022B #14 grade 11+ geometry-2d, algebra
The graph of y=x²+2x-15 intersects the x-axis at points A and C and the y-axis at point B. What is tan(∠ ABC)?
AMC 12 2022B #15 grade 8+ number-theory
One of the following numbers is not divisible by any prime number less than 10. Which is it?
AMC 12 2022B #16 grade 11+ algebra
Suppose x and y are positive real numbers such that x^y=2⁶⁴ and (log₂x)^log₂y=2⁷. What is the greatest possible value of…
AMC 12 2022B #17 grade 7+ counting
How many 4 × 4 arrays whose entries are 0s and 1s are there such that the row sums (the sum of the entries in each row)…
AMC 12 2022B #18 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 12 2022B #19 grade 11+ geometry-2d
In △ABC medians AD and BE intersect at G and △AGE is equilateral. Then cos(C) can be written as m√p/n, where m and n are…
AMC 12 2022B #20 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 12 2022B #21 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 12 2022B #22 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 12 2022B #23 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_…
AMC 12 2022B #24 grade 12+ geometry-2d, algebra
The figure below depicts a regular 7-gon inscribed in a unit circle. What is the sum of the 4th powers of the lengths of…
AMC 12 2022B #25 grade 8+ geometry-2d
Four regular hexagons surround a square with side length 1, each one sharing an edge with the square, as shown in the fi…

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