|
|
| Line 1: |
Line 1: |
| | __TOC__ | | __TOC__ |
| − | == Day 1 ==
| + | stop it. seriously. |
| | | | |
| − | === Problem 1 ===
| + | 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 22:39, 19 March 2024
