Question 36
Let $a, b, c$ positive real numbers such that $a + b + c = 1$. Prove that:
$$
(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \ge 8(ab+bc+ca).
$$
When does equality hold?
$$
(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \ge 8(ab+bc+ca).
$$
When does equality hold?