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a_ix_i + x_{i+2 }\equiv 0 (mod 5) where a_i sequence, x_i \in N sequence

Source: KJMO 2007 p1

May 2, 2019

number theorynumber theory with sequencesInteger sequenceSequencesmultiple

Problem Statement

A sequence

a_1,a_2,...,a_{2007}

where

a_i \in\{2,3\}

for

i = 1,2,...,2007

and an integer sequence

x_1,x_2,...,x_{2007}

satisfies the following:

a_ix_i + x_{i+2 }\equiv 0

(

mod 5

) , where the indices are taken modulo

2007

. Prove that

x_1,x_2,...,x_{2007}

are all multiples of

5