Let $a,b,c{}$ be positive real numbers. Prove that \[108\cdot(ab+bc+ca)\leqslant(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})^4.\]
Source: Russian TST 2018, Day 10 P1 (Groups A & B)
Tags: algebra, inequalities
Let $a,b,c{}$ be positive real numbers. Prove that \[108\cdot(ab+bc+ca)\leqslant(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})^4.\]