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IMO ShortList 2003, number theory problem 1

Source: IMO ShortList 2003, number theory problem 1

October 4, 2004

modular arithmeticnumber theorySequenceDivisibilityIMO Shortlist

Problem Statement

Let

m

be a fixed integer greater than

1

. The sequence

x_0

,

x_1

,

x_2

,

\ldots

is defined as follows:

x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}

Find the greatest

k

for which the sequence contains

k

consecutive terms divisible by

m

.
Proposed by Marcin Kuczma, Poland

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