IMO shortlist 1983/9 solution

24th IMO 1983 shortlist

Problem 9

Let m be any integer and n any positive integer. Show that there is a polynomial p(x) with integral coefficients such that | p(x) - m/n | < 1/n² for all x in some interval of length 1/n.

Solution

Take p(x) = (m/n) (1 + (nx-1)^2k+1) and the interval to be [1/2n, 3/2n]. If we expand p(x) by the binomial theorem it is obvious that p(x) has integral coefficients. Also |p(x) - m/n| = |m/n| |nx - 1|^2k+1 ≤ |m/n| 1/2^2k+1 which < 1/n² for k sufficiently large.

24th IMO shortlist 1983