Question 23
Let $a$, $b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that
$$
\frac{a}{a+b^2} + \frac{b}{b+c^2} + \frac{c}{c+a^2} \le \frac{1}{4} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right). \quad (\text{Tonći Kokan})
$$
$$
\frac{a}{a+b^2} + \frac{b}{b+c^2} + \frac{c}{c+a^2} \le \frac{1}{4} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right). \quad (\text{Tonći Kokan})
$$