MathNet 2020 — Math Olympiad Vault — Part 04 Question 137

Prove that for all positive $a,b,c,d$,
$$\sqrt{\frac a{b+c+d}}+\sqrt{\frac b{c+d+a}}+\sqrt{\frac c{d+a+b}}+\sqrt{\frac d{a+b+c}}\gt 2.$$