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Square of rational number

Source: Chinese TST 2009 P4

April 4, 2009
combinatorics proposedcombinatorics

Problem Statement

Let positive real numbers a,b a,b satisfy b \minus{} a > 2. Prove that for any two distinct integers m,n m,n belonging to [a,b), [a,b), there always exists non-empty set S S consisting of certain integers belonging to [ab,(a \plus{} 1)(b \plus{} 1)) such that ∏x∈Smn \frac {\displaystyle\prod_{x\in S}}{mn} is square of a rational number.
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