IMO Shortlist 1969 problem 18


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(FRA 1) Let a and b be two nonnegative integers. Denote by H(a, b) the set of numbers n of the form n = pa + qb, where p and q are positive integers. Determine H(a) = H(a, a). Prove that if a \neq b, it is enough to know all the sets H(a, b) for coprime numbers a, b in order to know all the sets H(a, b). Prove that in the case of coprime numbers a and b, H(a, b) contains all numbers greater than or equal to \omega = (a - 1)(b -1) and also \frac{\omega}{2} numbers smaller than \omega
Izvor: Međunarodna matematička olimpijada, shortlist 1969