IMO 1983 SL 10

Let p and q be integers. Show that there exists an interval I of

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IMO 1983 SL 10

Origin: FIN

Problem

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 P(x) −p q < 1 q2 for all x \inI.

Solution

Choose P(x) = p q  (qx −1)2n+1 + 1  , I = [1/2q, 3/2q]. Then all the coef- ficients of P are integers, and P(x) −p q

p q (qx −1)2n+1 \leq

p q

22n+1 , for x \inI. The desired inequality follows if n is chosen large enough.