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2025 SSMO Accuracy Round Problems/Problem 2

Revision as of 11:26, 9 September 2025 by Pinkpig (talk | contribs) (Created page with "==Problem== Let <math>ABC</math> be a triangle with circumcircle <math>\omega</math>. The midpoint of <math>AB</math> is <math>M,</math> and the line <math>CM</math> intersec...")
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Problem

Let $ABC$ be a triangle with circumcircle $\omega$. The midpoint of $AB$ is $M,$ and the line $CM$ intersects $\omega$ again at $P$. Given $\angle BMC = 120^\circ,$ $\triangle BMC$ is isosceles, and $BC = 20,$ the length of $PM$ can be written as $\tfrac{a\sqrt{b}}{c},$ where $a,$ $b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is square-free. Find $a+b+c$.

Solution

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