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Prove the upper bound on the sequence x_n

Source: Balkan MO ShortList 2009 A3

April 6, 2020

Problem Statement

Denote by

S(x)

the sum of digits of positive integer

x

written in decimal notation. For

k

a fixed positive integer, define a sequence

x_{n+1}

by

x_1=1

and

x_{n+1}

=

S(kx_n)

for all positive integers

n

. Prove that

x_n

<

27 \sqrt{k}

for all positive integer

n

.

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