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Difference between revisions of "2021 USAMO Problems/Problem 3"

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A perfect number is a positive integer that is equal to the sum of its proper divisors, such as <math>6</math>, <math>28</math>, <math>496</math>, and <math>8,128</math>. Prove that
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Let <math>n \geq 2</math> be an integer. An <math>n \times n</math> board is initially empty. Each minute, you may perform one of three moves:
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*If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells.
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(1) All even perfect numbers follow the format <math>\frac{1}{2}M(M+1)</math>, where <math>M</math> is a Mersenne prime;
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*If all cells in a column have a stone, you may remove all stones from that column.
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*If all cells in a row have a stone, you may remove all stones from that row.
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(2) All <math>\frac{1}{2}M*(M+1)</math>, where <math>M</math> is a Mersenne prime, are even perfect numbers;
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<asy>
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unitsize(20);
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(3) There are no odd perfect numbers.
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draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0));
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fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey);
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Note: a Mersenne prime is a prime in the form of <math>2^p-1</math>.
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draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2));
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draw((0,2)--(4,2));
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draw((2,4)--(2,0));
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</asy>
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For which <math>n</math> is it possible that, after some non-zero number of moves, the board has no stones?
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== Solution (WIP) ==

Latest revision as of 12:42, 25 December 2023

Let $n \geq 2$ be an integer. An $n \times n$ board is initially empty. Each minute, you may perform one of three moves:

  • If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells.
  • If all cells in a column have a stone, you may remove all stones from that column.
  • If all cells in a row have a stone, you may remove all stones from that row.

[asy] unitsize(20); draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)); fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey); draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2)); draw((0,2)--(4,2)); draw((2,4)--(2,0)); [/asy] For which $n$ is it possible that, after some non-zero number of moves, the board has no stones?

Solution (WIP)

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