Question 44
Let $a, b, c$ be real numbers such that $0 \lt a, b, c \lt 1/2$ and $a + b + c = 1$. Prove that for all real numbers $x, y, z$,
$$
abc(x + y + z)^2 \geq ayz(1 - 2a) + bxz(1 - 2b) + cxy(1 - 2c).
$$
When does equality hold?
$$
abc(x + y + z)^2 \geq ayz(1 - 2a) + bxz(1 - 2b) + cxy(1 - 2c).
$$
When does equality hold?