Question 153
Let $a$, $b$, $c \gt 0$. Prove that
$$
\frac{a+b}{a^2+b^2} + \frac{b+c}{b^2+c^2} + \frac{c+a}{c^2+a^2} \le \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.
$$
$$
\frac{a+b}{a^2+b^2} + \frac{b+c}{b^2+c^2} + \frac{c+a}{c^2+a^2} \le \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.
$$