Question 197
Suppose that $a, b, c$ are positive real numbers, prove that
$$
1 \lt \frac{a}{\sqrt{a^2 + b^2}} + \frac{b}{\sqrt{b^2 + c^2}} + \frac{c}{\sqrt{c^2 + a^2}} \le \frac{3\sqrt{2}}{2}.
$$
$$
1 \lt \frac{a}{\sqrt{a^2 + b^2}} + \frac{b}{\sqrt{b^2 + c^2}} + \frac{c}{\sqrt{c^2 + a^2}} \le \frac{3\sqrt{2}}{2}.
$$