IMO 1985 LL CZS23

Let N = {1, 2, 3, . . .}. For real x, y, set S(x, y) = {s | s =

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IMO 1985 LL CZS23

Origin: CZS

Problem

Let N = {1, 2, 3, . . .}. For real x, y, set S(x, y) = {s | s = [nx + y], n \inN}. Prove that if r > 1 is a rational number, there exist real numbers u and v such that S(r, 0) \capS(u, v) = \emptyset, S(r, 0) \cupS(u, v) = N.