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There exists an interval I

Source: IMO Longlist 1983, Problem 23

September 9, 2010

algebrapolynomialapproximationnumber theoryrational numberIMO Shortlist

Problem Statement

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.