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Rectangle that is the disjoint union of a finite number of r

Source: IMO Shortlist 1989, Problem 8, ILL 20

September 18, 2008
geometryrectanglecalculusintegrationnumber theoryIMO Shortlist

Problem Statement

Let R R be a rectangle that is the union of a finite number of rectangles Ri, R_i, 1≤i≤n, 1 \leq i \leq n, satisfying the following conditions: (i) The sides of every rectangle Ri R_i are parallel to the sides of R. R. (ii) The interiors of any two different rectangles Ri R_i are disjoint. (iii) Each rectangle Ri R_i has at least one side of integral length. Prove that R R has at least one side of integral length. Variant: Same problem but with rectangular parallelepipeds having at least one integral side.
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