MathDB
Problems
Contests
International Contests
Middle European Mathematical Olympiad
2022 Middle European Mathematical Olympiad
8
8
Part of
2022 Middle European Mathematical Olympiad
Problems
(1)
Obtain the AM of the digits by putting a decimal separator
Source: MEMO 2022 T8
9/2/2022
We call a positive integer
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
h
e
e
s
y
<
/
s
p
a
n
>
<span class='latex-italic'>cheesy</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
h
eesy
<
/
s
p
an
>
if we can obtain the average of the digits in its decimal representation by putting a decimal separator after the leftmost digit. Prove that there are only finitely many
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
h
e
e
s
y
<
/
s
p
a
n
>
<span class='latex-italic'>cheesy</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
h
eesy
<
/
s
p
an
>
numbers.
number theory