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Difference between revisions of "2022 USAJMO Problems"

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[[2022 USAJMO Problems/Problem 1|Solution]]
 
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===Problem 2===
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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.
  
 
[[2022 USAJMO Problems/Problem 2|Solution]]
 
[[2022 USAJMO Problems/Problem 2|Solution]]
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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.
 
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===Problem 3===
 
===Problem 3===
  

Revision as of 20:10, 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

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.

Solution

Problem 3

Solution

Day 2

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

2021 USAJMO (Problems • Resources)
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All USAJMO Problems and Solutions

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