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Is it possible to solve this by generating functions?

Source: Romanian IMO Team Selection Test TST 1991, problem 14

May 24, 2005
functioncombinatorics proposedcombinatorics

Problem Statement

Let a1<a2<⋯<ana_1<a_2<\cdots<a_n be positive integers. Some colouring of Z\mathbb{Z} is periodic with period tt such that for each x∈Zx\in \mathbb{Z} exactly one of x+a1,x+a2,…,x+anx+a_1,x+a_2,\dots,x+a_n is coloured. Prove that n∣tn\mid t. Andrei Radulescu-Banu
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