Problem

Source: 2017 Romania JBMO TST 2.4

Tags: inequalities, algebra



Let $a, b, c, d$ be non-negative real numbers satisfying $a + b + c + d = 3$. Prove that $$\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} +\frac{c}{1 + 2d^3} +\frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3}$$When does the equality hold?