Let and be integers. Show that there exists an interval of length and a polynomial with integral coefficients such that for all
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Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that
$$\left|P(x)-\frac pq \right| < \frac{1}{q^2}$$
for all $x \in I.$
Izvor: Međunarodna matematička olimpijada, shortlist 1983
Školjka 
and 
be integers. Show that there exists an interval 
of length 
and a polynomial 
with integral coefficients such that
