KöMaL 2023 B5356

Prove the following inequality for every integer $n\geq 2$ and non-negative real numbers $x_{1},\ldots,x_{n}$: $\sqrt[\scriptstyle n]{\prod\limits_{i=1}^{n} (1+x_{i})}\ge 1+\sqrt[\scriptstyle n]{\prod\limits_{i=1}^{n}x_{i}}.$