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2024 SSMO Team Round Problems/Problem 4

Revision as of 15:44, 2 May 2025 by Pinkpig (talk | contribs) (Created page with "==Problem== Let <math>ABC</math> be a right triangle with circumcenter <math>O</math> and incenter <math>I</math> such that <math>\angle ABC = 90^{\circ}</math> and <math>\fr...")
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Problem

Let $ABC$ be a right triangle with circumcenter $O$ and incenter $I$ such that $\angle ABC = 90^{\circ}$ and $\frac{AB}{BC} = \frac{3}{4}$. Let $D$ the projection of $O$ onto $AB$, and let $E$ be the projection of $O$ onto $BC$. Denote $\omega_{1}$ be the incenter of $ADO$ and $\omega_{2}$ as the incenter of $OEC$. If $\frac{[\omega_{1}\omega_{2}I]}{[ABC]}=\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

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