SMO 2016 Problem 2

Let $a, b, c$ be real numbers such that $0 < a, b, c < 1/2$ and $a + b + c= 1$ . Prove that for all real numbers $x,y,z$ , $$abc(x + y + z)^2 \ge ayz( 1- 2a) + bxz( 1 - 2b) + cxy( 1 - 2c)$$ . When does equality hold?