Difference between revisions of "2022 SSMO Speed Round Problems/Problem 9"

Latest revision as of 19:14, 2 May 2025

Problem

Consider a triangle $ABC$ such that $AB=13$ , $BC=14$ , $CA=15$ and a square $WXYZ$ such that $Y$ and $Z$ lie on $\overleftrightarrow{BC}$ , $W$ lies on $\overleftrightarrow{AB}$ , and $X$ lies on $\overleftrightarrow{CA}$ . Suppose further that $W$ , $X$ , $Y$ , and $Z$ are distinct from $A$ , $B$ , and $C$ . Let $O$ be the center of $WXYZ$ . If $AO$ intersects $BC$ at $P$ , then the sum of all values of $\frac{BP}{CP}$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

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+==Problem==
+Consider a triangle <math>ABC</math> such that <math>AB=13</math>, <math>BC=14</math>, <math>CA=15</math> and a square <math>WXYZ</math> such that <math>Y</math> and <math>Z</math> lie on <math>\overleftrightarrow{BC}</math>, <math>W</math> lies on <math>\overleftrightarrow{AB}</math>, and <math>X</math> lies on <math>\overleftrightarrow{CA}</math>. Suppose further that <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> are distinct from <math>A</math>, <math>B</math>, and <math>C</math>. Let <math>O</math> be the center of <math>WXYZ</math>. If <math>AO</math> intersects <math>BC</math> at <math>P</math>, then the sum of all values of <math>\frac{BP}{CP}</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+==Solution==

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Difference between revisions of "2022 SSMO Speed Round Problems/Problem 9"

Latest revision as of 19:14, 2 May 2025

Problem

Solution