Difference between revisions of "2024 USAJMO Problems"

Revision as of 21:33, 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 3 ===		=== 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$.	+	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>.

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