Problem

Source: Stars of Mathematics 2018, Juniors, Problem 4

Tags: inequalities



Let be a natural number $ n\ge 4 $ and $ n $ nonnegative numbers $ a,b,\ldots ,c. $ Prove that $$ \prod_{\text{cyc} } (a+b+c)^2 \ge 2^n\prod_{\text{cyc} } (a+b)^2, $$and tell in which circumstances equality happens.