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Problem 914

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Triangles inside Circles

For a given integer $R$ consider all primitive Pythagorean triangles that can fit inside, without touching, a circle with radius $R$. Define $F(R)$ to be the largest inradius of those triangles. You are given $F(100)=36$.

Find $F({10}^{18})$.

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圆内三角形

对于给定的整数$R$，考虑所有可以完全放入半径为$R$的圆内，且不接触圆周的本原毕达哥拉斯三角形。定义$F(R)$为这些三角形中最大的内切圆半径。已知$F(100)=36$。

求$F({10}^{18})$。

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