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Balkan MO Shortlist
2009 Balkan MO Shortlist
A3
A3
Part of
2009 Balkan MO Shortlist
Problems
(1)
Prove the upper bound on the sequence x_n
Source: Balkan MO ShortList 2009 A3
4/6/2020
Denote by
S
(
x
)
S(x)
S
(
x
)
the sum of digits of positive integer
x
x
x
written in decimal notation. For
k
k
k
a fixed positive integer, define a sequence
(
x
n
)
n
≥
1
(x_n)_{n \geq 1}
(
x
n
)
n
≥
1
by
x
1
=
1
x_1=1
x
1
=
1
and
x
n
+
1
x_{n+1}
x
n
+
1
=
=
=
S
(
k
x
n
)
S(kx_n)
S
(
k
x
n
)
for all positive integers
n
n
n
. Prove that
x
n
x_n
x
n
<
<
<
27
k
27 \sqrt{k}
27
k
for all positive integer
n
n
n
.