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Expensive n-tuples

Source: EGMO 2017 P5

April 9, 2017
number theoryequationEGMOEGMO 2017

Problem Statement

Let n≥2n\geq2 be an integer. An nn-tuple (a1,a2,…,an)(a_1,a_2,\dots,a_n) of not necessarily different positive integers is expensive if there exists a positive integer kk such that (a1+a2)(a2+a3)…(an−1+an)(an+a1)=22k−1.(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}. a) Find all integers n≥2n\geq2 for which there exists an expensive nn-tuple.
b) Prove that for every odd positive integer mm there exists an integer n≥2n\geq2 such that mm belongs to an expensive nn-tuple.
There are exactly nn factors in the product on the left hand side.
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