TW-TST 1999 Problem 2

Let $a_{1},a_{2},...,a_{1999}$ be a sequence of nonnegative integers such that for any $i,j$ with $i+j\leq 1999$ , $a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1$ . Prove that there exists a real number $x$ such that $a_{n}=[nx]\forall n$ .