2024 USAJMO Problems/Problem 1

Problem

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 1 (One-liner)

$OP=OQ=\sqrt{R^2-3.5^2+0.5^2}=\sqrt{R^2-12}=\sqrt{R^2-4^2+2^2}=OR=OS$

2024 USAJMO (Problems • Resources)
Preceded by First Question	Followed by Problem 2
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2024 USAJMO Problems/Problem 1

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Problem

Solution 1 (One-liner)

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