2024 USAJMO Problems

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.

Solution

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.

Solution

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 $p^k$ .

Solution

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2024 USAJMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3