Math Gold Medalist

Lor

2023 AIME II 

Problem 9

Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q,$ and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B,$ respectively. The line parallel to $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y,$ respectively. Suppose $PX=10,$ $PY=14,$ and $PQ=5.$ Then the area of trapezoid $XABY$ is $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n.$

Power of a Point

Radical Axis

Perpendicular Bisector

Drawing Perpendiculars

Pythagorean Theorem

.

Extend PQ to intersect AB at point C

CA=CB=6

Consider centers of w1 and w2 are O1 and O2

Draw O1A and O2B

CP=4

Then use Pythagorean theorem to find height of trapezoid

   Solution