Question 48
Let $a, b, c, d$ be non-negative real numbers such that $a + b + c + d = 4$. Prove that
$$
a\sqrt{3a+b+c}+b\sqrt{3b+c+d}+c\sqrt{3c+d+a}+d\sqrt{3d+a+b} \ge 4\sqrt{5}.
$$
$$
a\sqrt{3a+b+c}+b\sqrt{3b+c+d}+c\sqrt{3c+d+a}+d\sqrt{3d+a+b} \ge 4\sqrt{5}.
$$