Nemotron-IMO-Bench 2026 A34

A sequence $(x_n)_{n\geq 1}$ of real numbers has the property that $x_1>0$ and that for all $n\geq 1$ $$x_{n+1}\geq (n+2)x_n-\sum_{k=1}^{n-1} kx_k.$$ Prove that $x_n\geq n!x_1$ for all $n\geq 1$.