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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2022 Poland - Second Round
5
5
Part of
2022 Poland - Second Round
Problems
(1)
Polish MO 2022 P5
Source: Polish MO 2022 P5
2/12/2022
Let
n
n
n
be an positive integer. We call
n
n
n
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
when there exists positive integer
k
k
k
s.t.
n
=
k
(
k
+
1
)
n=k(k+1)
n
=
k
(
k
+
1
)
. Does there exist 2022 pairwise distinct
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
numbers s.t. their sum is also
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
number?
algebra
number theory