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IMO ShortList 2002, combinatorics problem 6

Source: IMO ShortList 2002, combinatorics problem 6; 54th Polish 2003

September 28, 2004
graph theorycombinatoricsEulerian pathIMO Shortlist

Problem Statement

Let nn be an even positive integer. Show that there is a permutation (x1,x2,…,xn)\left(x_{1},x_{2},\ldots,x_{n}\right) of (1, 2, …,n)\left(1,\,2,\,\ldots,n\right) such that for every i∈{1, 2, ..., n}i\in\left\{1,\ 2,\ ...,\ n\right\}, the number xi+1x_{i+1} is one of the numbers 2xi2x_{i}, 2xi−12x_{i}-1, 2xi−n2x_{i}-n, 2xi−n−12x_{i}-n-1. Hereby, we use the cyclic subscript convention, so that xn+1x_{n+1} means x1x_{1}.
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