2019 AIME I Problems/Problem 15

Problem 15

Let $\overline{AB}$ be a chord of a circle $\omega$ , and let $P$ be a point on the chord $\overline{AB}$ . Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$ . Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$ . Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$ . Line $PQ$ intersects $\omega$ at $X$ and $Y$ . Assume that $AP=5$ , $PB=3$ , $XY=11$ , and $PQ^2 = \tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

2019 AIME I (Problems • Answer Key • Resources)
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2019 AIME I Problems/Problem 15

Problem 15

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