RO-TST 1999 Problem 8

Let $a$ be a positive real number and $\{x_n\}_{n\geq 1}$ a sequence of real numbers such that $x_1=a$ and $$x_{n+1} \geq (n+2)x_n - \sum^{n-1}_{k=1}kx_k, \ \forall \ n\geq 1.$$ Prove that there exists a positive integer $n$ such that $x_n > 1999!$ .

Ciprian Manolescu