Problem

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Tags: inequalities, geometry, Cauchy Inequality, inequalities unsolved



$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that \[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]