Question 117
a) $\left(\frac{2a}{b+c}\right)^2 + \left(\frac{2b}{c+a}\right)^2 + \left(\frac{2c}{a+b}\right)^2 \ge \frac{a}{c} + \frac{b}{a} + \frac{c}{b}$.
b) $\left(\frac{a+b}{c}\right)^2 + \left(\frac{b+c}{a}\right)^2 + \left(\frac{c+a}{b}\right)^2 \ge \frac{a}{b} + \frac{b}{c} + \frac{c}{a} + 9$.
For each of them either prove that it holds true for all positive real numbers $a, b$, and $c$, or provide a triple $(a, b, c) \in \mathbb{R}^{+}$, which does not satisfy it.
b) $\left(\frac{a+b}{c}\right)^2 + \left(\frac{b+c}{a}\right)^2 + \left(\frac{c+a}{b}\right)^2 \ge \frac{a}{b} + \frac{b}{c} + \frac{c}{a} + 9$.
For each of them either prove that it holds true for all positive real numbers $a, b$, and $c$, or provide a triple $(a, b, c) \in \mathbb{R}^{+}$, which does not satisfy it.