Let $a, b, c$ be positive real numbers. Prove that $$\frac{a(b^2 + c^2)}{(b + c)(a^2 + bc)} + \frac{b(c^2 + a^2)}{(c + a)(b^2 + ca)} + \frac{c(a^2 + b^2)}{(a + b)(c^2 + ab)} \ge \frac32$$
Source: 2017 Saudi Arabia Mock BMO II p1
Tags: inequalities, algebra
Let $a, b, c$ be positive real numbers. Prove that $$\frac{a(b^2 + c^2)}{(b + c)(a^2 + bc)} + \frac{b(c^2 + a^2)}{(c + a)(b^2 + ca)} + \frac{c(a^2 + b^2)}{(a + b)(c^2 + ab)} \ge \frac32$$