2024 USAJMO Problems: Difference between revisions

Revision as of 20:33, 19 March 2024

Day 1

Problem 1

Let $ABCD$ be a cyclic quadrilateral with $AB=7$ and $CD=8$ . Points $P$ and $Q$ are selected on line segment $AB$ so that $AP=BQ=3$ . Points $R$ and $S$ are selected on line segment $CD$ so that $CR=DS=2$ . Prove that $PQRS$ is a quadrilateral.

Problem 2

Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x,y)$ with $1\leq x\leq2m$ and $1\leq y\leq2n$ . A configuration of $mn$ rectangles is called happy if each point in $S$ 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 $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1~ for each integer$ n\geq1 $. Suppose that$ p>2 $is prime and$ k $is a positive integer. Prove that some term of the sequence$ a(n) $is divisible by$ p^k$.

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 === 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 ===
-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'' uf 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.
+=== 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$.

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