Estonian Math Competitions 2005/2006
Source: Seniors Problem 6
July 30, 2008
rotationpigeonhole principlegeometrygeometric transformationcombinatorics unsolvedcombinatorics
Problem Statement
Kati cut two equal regular n\minus{}gons out of paper. To the vertices of both n\minus{}gons, she wrote the numbers 1 to in some order. Then she stabbed a needle through the centres of these n\minus{}gons so that they could be rotated with respect to each other. Kati noticed that there is a position where the numbers at each pair of aligned vertices are different. Prove that the n\minus{}gons can be rotated to a position where at least two pairs of aligned vertices contain equal numbers.