Difference between revisions of "2022 USAJMO Problems"
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[[2022 USAJMO Problems/Problem 2|Solution]] | [[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 | + | 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. |
| − | 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=== | ===Problem 3=== | ||
Revision as of 20:08, 19 April 2022
Contents
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 .
Problem 2
Let 
and 
be positive integers. The cells of an 
grid are colored amber and bronze such that there are at least $a^2 + ab − b$ (Error compiling LaTeX. Unknown error_msg) amber cells and at least $b^2 + ab − a$ (Error compiling LaTeX. Unknown error_msg) 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.
Problem 3
Day 2
Problem 4
Problem 5
Problem 6
| 2021 USAJMO (Problems • Resources) | ||
| Preceded by 2021 USAJMO |
Followed by 2023 USAJMO | |
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| All USAJMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
