Inequality
For $a,b,c>0 , a+b+c=1$ prove that
$$ \frac{a^2}{1+(a+b)^2}+\frac{b^2}{1+(b+c)^2}+\frac{c^2}{1+(c+a)^2}\le\frac{2187}{13}\cdot\frac{a^8+b^8+c^8}{ab+bc+ca} $$
Functional Equation
Determine all continuous function $ f\colon\mathbb{(0,+\infty)} \to\mathbb{(0,+\infty)} $ satisfying
$$ f(x^3)+f(y^3)+f(z^3)=f(xyz)f(\frac{x}{y})f(\frac{y}{z})f(\frac{z}{x}) $$
Geometry
Let $l$ be a tangent to the incircle of triangle $ABC$. Let $l_{a},l_{b}$ and $l_{c}$ be the respective images
of $l$ under reflection across the exterior bisector of $\hat A,\hat B$and $\hat C$. Prove that the triangle formed by these lines is congruent to $ABC$.
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