2023 WSMO Team Round Problems/Problem 7

Problem

In triangle $ABC$ with $AB = 13,AC = 14$ , and $BC = 15,$ a rectangle $WXYZ$ is inscribed such that the area of $WXYZ$ is maximized. If the minimum possible value of $\frac{WX}{XY}$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

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2023 WSMO Team Round Problems/Problem 7

Problem

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