USAMO 1997 Problem 6

Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies $$a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1$$ for all $i,j \geq 1$ with $i + j \leq 1997$ . Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$ ) for all $1 \leq n \leq 1997$ .