Difference between revisions of "2024 USAJMO Problems"

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== Day 1 ==
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stop it. seriously.
  
=== Problem 1 ===
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DO NOT PUT ANYTHING HERE UNTIL AFTER DISCUSSION PERMITTED PLEASE
Let <math>ABCD</math> be a cyclic quadrilateral with <math>AB=7</math> and <math>CD=8</math>. Points <math>P</math> and <math>Q</math> are selected on line segment <math>AB</math> so that <math>AP=BQ=3</math>. Points <math>R</math> and <math>S</math> are selected on line segment <math>CD</math> so that <math>CR=DS=2</math>. Prove that <math>PQRS</math> is a quadrilateral.
 
 
 
[[2024 USAJMO Problems/Problem 1|Solution]]
 
 
 
=== Problem 2 ===
 
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.
 
 
 
[[2024 USAJMO Problems/Problem 2|Solution]]
 
 
 
=== Problem 3 ===
 
Let <math>a(n)</math> be the sequence defined by <math>a(1)=2</math> and <math>a(n+1)=(a(n))^{n+1}-1</math> for each integer <math>n\geq1</math>. Suppose that <math>p>2</math> is prime and <math>k</math> is a positive integer. Prove that some term of the sequence <math>a(n)</math> is divisible by <math>p^k</math>.
 
 
 
[[2024 USAJMO Problems/Problem 3|Solution]]
 
  
 
== See also ==
 
== See also ==

Revision as of 21:39, 19 March 2024

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stop it. seriously.

DO NOT PUT ANYTHING HERE UNTIL AFTER DISCUSSION PERMITTED PLEASE

See also

2024 USAJMO (ProblemsResources)
Preceded by
2023 USAJMO Problems
Followed by
2025 USAJMO Problems
1 2 3 4 5 6
All USAJMO Problems and Solutions

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