Question 24
Prove that
$$
\frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)} + \frac{(c+1)(a+2)}{(a+1)(a+5)} \ge \frac{3}{2}
$$
for all positive real numbers $a$, $b$, $c$ satisfying the condition $a^2 + b^2 + c^2 \ge 3$.
$$
\frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)} + \frac{(c+1)(a+2)}{(a+1)(a+5)} \ge \frac{3}{2}
$$
for all positive real numbers $a$, $b$, $c$ satisfying the condition $a^2 + b^2 + c^2 \ge 3$.