58th Ukrainian National Mathematical Olympiad

Question

Determine all possible pairs of integers a, b so that exactly one of them is even and so that there are non-integer x, y, such that both x+y and ax+by are integer?

Accepted Answer

Since ax+by = a(x+y) + (b-a)y, the value of (b-a)y is integer. If |a-b|=1, then y has to be integer – contradiction. For |a-b|>1 we can let y = 1b-a and x = -y. Then x+y=0, (b-a)y is integer, hence ax+by is also integer.