USAMO 1997 — Days 1 and 2 Question 6

Nonnegative integers $a_1,a_2,\ldots,a_{1997}$ satisfy $$a_i+a_j\le a_{i+j}\le a_i+a_j+1$$ for all $i,j\ge1$ with $i+j\le1997$. Prove that there exists a real number $x$ such that $a_n=\lfloor nx\rfloor$ for every $1\le n\le1997$.