USAMO 1983 Problem 5

Prove that an open interval of length $1 / \mathrm{n}$ in the real line contains at most $(\mathrm{n}+1) / 2$ rational points $\mathrm{p} / \mathrm{q}$ with $1 \leq$ $\mathrm{q} \leq \mathrm{n}$.