MathDB
IMO LongList 1985 CZS4 - Prove That u,v exist

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September 10, 2010
floor functioninequalitiesnumber theoryrelatively primenumber theory proposed

Problem Statement

Let N=1,2,3,...\mathbb N = {1, 2, 3, . . .}. For real x,yx, y, set S(x,y)={s∣s=[nx+y],n∈N}S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}. Prove that if r>1r > 1 is a rational number, there exist real numbers uu and vv such that S(r,0)∩S(u,v)=∅,S(r,0)∪S(u,v)=N.S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.
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