Difference between revisions of "2024 USAJMO Problems"
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=== Problem 1 === | === Problem 1 === | ||
− | 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 | + | 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. |
=== Problem 2 === | === 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'' | + | 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. |
+ | |||
+ | === 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~ 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$. |
Revision as of 20:33, 19 March 2024
Contents
[hide]Day 1
Problem 1
Let be a cyclic quadrilateral with and . Points and are selected on line segment so that . Points and are selected on line segment so that . Prove that is a quadrilateral.
Problem 2
Let and be positive integers. Let be the set of integer points with and . A configuration of rectangles is called happy if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Problem 3
Let be the sequence defined by and n\geq1p>2ka(n)p^k$.