Difference between revisions of "2022 USAJMO Problems"

Revision as of 20:09, 19 April 2022

Day 1

$\textbf{Note:}$ 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 $m$ does there exist an infinite arithmetic sequence of integers $a_1,a_2,\cdots$ and an infinite geometric sequence of integers $g_1,g_2,\cdots$ satisfying the following properties?

$\bullet$ $a_n-g_n$ is divisible by $m$ for all integers $n>1$ ;

$\bullet$ $a_2-a_1$ is not divisible by $m$ .

Solution

Problem 2

Solution

Let $a$ and $b$ be positive integers. The cells of an $(a + b + 1)\times (a + b + 1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

Problem 3

Solution

Day 2

2021 USAJMO (Problems • Resources)
Preceded by 2021 USAJMO	Followed by 2023 USAJMO
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 [[2022 USAJMO Problems/Problem 2|Solution]]
-Let <math>a</math> and <math>b</math> be positive integers. The cells of an <math>(a + b + 1)\times (a + b + 1)</math> grid are colored amber and bronze such that there are at least <math>a^2 + ab − b</math> amber cells and at least <math>b^2 + ab − a</math> bronze cells. Prove that it is possible to choose <math>a</math> amber cells and <math>b</math> bronze cells such that no two of the <math>a + b</math> chosen cells lie in the same row or column.
+Let <math>a</math> and <math>b</math> be positive integers. The cells of an <math>(a + b + 1)\times (a + b + 1)</math> grid are colored amber and bronze such that there are at least <math>a^2+ab-b</math> amber cells and at least <math>b^2+ab-a</math> bronze cells. Prove that it is possible to choose <math>a</math> amber cells and <math>b</math> bronze cells such that no two of the <math>a+b</math> chosen cells lie in the same row or column.
 ===Problem 3===

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Difference between revisions of "2022 USAJMO Problems"

Revision as of 20:09, 19 April 2022

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6