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Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2021 Philippine MO
2
pmo problem 2
pmo problem 2
Source: PMO
March 19, 2021
algebra
number theory
PMO
Problem Statement
Let
n
n
n
be a positive integer. Show that there exists a one-to-one function
σ
:
{
1
,
2
,
.
.
.
,
n
}
→
{
1
,
2
,
.
.
.
,
n
}
\sigma : \{1,2,...,n\} \to \{1,2,...,n\}
σ
:
{
1
,
2
,
...
,
n
}
→
{
1
,
2
,
...
,
n
}
such that
∑
k
=
1
n
k
(
k
+
σ
(
k
)
)
2
<
1
2
.
\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^2} < \frac{1}{2}.
k
=
1
∑
n
(
k
+
σ
(
k
)
)
2
k
<
2
1
.
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