Difference between revisions of "2024 USAJMO Problems/Problem 2"
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Let <math>m</math> and <math>n</math> be positive integers. Let <math>S</math> be the set of integer points <math>(x,y)</math> with <math>1\leq x\leq2m</math> and <math>1\leq y\leq2n</math>. A configuration of <math>mn</math> rectangles is called ''happy'' if each point in <math>S</math> is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd. | Let <math>m</math> and <math>n</math> be positive integers. Let <math>S</math> be the set of integer points <math>(x,y)</math> with <math>1\leq x\leq2m</math> and <math>1\leq y\leq2n</math>. A configuration of <math>mn</math> rectangles is called ''happy'' if each point in <math>S</math> is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd. | ||
| − | == Solution 1 == | + | |
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| + | === Solution 1 === | ||
==See Also== | ==See Also== | ||
{{USAJMO newbox|year=2024|num-b=1|num-a=3}} | {{USAJMO newbox|year=2024|num-b=1|num-a=3}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 20:38, 20 March 2024
Contents
Problem
Let 
and 
be positive integers. Let 
be the set of integer points 
with 
and 
. A configuration of 
rectangles is called happy if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Solution 1
See Also
| 2024 USAJMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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. 
