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Problems
Contests
National and Regional Contests
Peru Contests
Peru Cono Sur TST
2021 Peru Cono Sur TST.
P2
P2
Part of
2021 Peru Cono Sur TST.
Problems
(1)
Fascinating numbers
Source: 2021 Peru Cono Sur TST P2
7/11/2023
For each positive integer
k
k
k
we denote by
S
(
k
)
S(k)
S
(
k
)
the sum of its digits, for example
S
(
132
)
=
6
S(132)=6
S
(
132
)
=
6
and
S
(
1000
)
=
1
S(1000)=1
S
(
1000
)
=
1
. A positive integer
n
n
n
is said to be
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
s
c
i
n
a
t
i
n
g
<
/
s
p
a
n
>
<span class='latex-bold'>fascinating</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
sc
ina
t
in
g
<
/
s
p
an
>
if it holds that
n
=
k
S
(
k
)
n = \frac{k}{S(k)}
n
=
S
(
k
)
k
for some positive integer
k
k
k
. For example, the number
11
11
11
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
s
c
i
n
a
t
i
n
g
<
/
s
p
a
n
>
<span class='latex-bold'>fascinating</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
sc
ina
t
in
g
<
/
s
p
an
>
since
11
=
198
S
(
198
)
(
11 = \frac{198}{S(198)} (
11
=
S
(
198
)
198
(
since
198
S
(
198
)
=
198
1
+
9
+
8
=
198
18
=
11
)
\frac{198}{S(198)}=\frac{198}{1+9+8}=\frac{198}{18} = 11)
S
(
198
)
198
=
1
+
9
+
8
198
=
18
198
=
11
)
. Prove that there exists a positive integer less than
2021
2021
2021
and that it is not
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
s
c
i
n
a
t
i
n
g
<
/
s
p
a
n
>
<span class='latex-bold'>fascinating</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
sc
ina
t
in
g
<
/
s
p
an
>
.
number theory