2022 USAJMO Problems

Day 1

For any geometry problem whose statement begins with an asterisk , the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 1

For which positive integers does there exist an infinite arithmetic sequence of integers and an infinite geometric sequence of integers satisfying the following properties?

is divisible by for all integers ;

is not divisible by .

Solution

Problem 2

Let and be positive integers. The cells of an grid are colored amber and bronze such that there are at least amber cells and at least bronze cells. Prove that it is possible to choose amber cells and bronze cells such that no two of the chosen cells lie in the same row or column.

Solution

Problem 3

Solution Let and be fixed integers, and . Given are identical black rods and identical white rods, each of side length 1.

We assemble a regular -gon using these rods so that parallel sides are the same color. Then, a convex -gon is formed by translating the black rods, and a convex -gon is formed by translating the white rods. An example of one way of doing the assembly when and is shown below, as well as the resulting polygons and .

[image here]

Prove that the difference of the areas of and depends only on the numbers and , and not on how the -gon was assembled.

Day 2

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Let be complex numbers, and define \begin{align*} a_{n+1}&=a_n^2+2b_nc_n\\ b_{n+1}&=b_n^2+2c_na_n\\ c_{n+1}&=c_n^2+2a_nb_n \end{align*} for all nonnegative integers .

Suppose that for all . Prove that

2021 USAJMO (Problems • Resources)
Preceded by 2021 USAJMO	Followed by 2023 USAJMO
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.