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239 Open Math Olympiad
2017 239 Open Mathematical Olympiad
1
An Average on Permutations
An Average on Permutations
Source: 239 2017 J1
May 31, 2020
combinatorics
permutations
Problem Statement
Denote every permutation of
1
,
2
,
…
,
n
1,2,\dots, n
1
,
2
,
…
,
n
as
σ
=
(
a
1
,
a
2
,
…
,
n
)
\sigma =(a_1,a_2,\dots,n)
σ
=
(
a
1
,
a
2
,
…
,
n
)
. Prove that the sum
∑
1
(
a
1
)
(
a
1
+
a
2
)
(
a
1
+
a
2
+
a
3
)
…
(
a
1
+
a
2
+
⋯
+
a
n
)
\sum \frac{1}{(a_1)(a_1+a_2)(a_1+a_2+a_3)\dots(a_1+a_2+\dots+a_n)}
∑
(
a
1
)
(
a
1
+
a
2
)
(
a
1
+
a
2
+
a
3
)
…
(
a
1
+
a
2
+
⋯
+
a
n
)
1
taken over all possible permutations
σ
\sigma
σ
equals
1
n
!
\frac{1}{n!}
n
!
1
.
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