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2006 Costa Rica - Final Round
1
Find SUM 1/(i_1 i_2 ... i_r)
Find SUM 1/(i_1 i_2 ... i_r)
Source:
April 12, 2006
algebra proposed
algebra
Problem Statement
Consider the set
S
=
{
1
,
2
,
.
.
.
,
n
}
S=\{1,2,...,n\}
S
=
{
1
,
2
,
...
,
n
}
. For every
k
∈
S
k\in S
k
∈
S
, define
S
k
=
{
X
⊆
S
,
k
∉
X
,
X
≠
∅
}
S_{k}=\{X \subseteq S, \ k \notin X, X\neq \emptyset\}
S
k
=
{
X
⊆
S
,
k
∈
/
X
,
X
=
∅
}
. Determine the value of the sum
S
k
∗
=
∑
{
i
1
,
i
2
,
.
.
.
,
i
r
}
∈
S
k
1
i
1
⋅
i
2
⋅
.
.
.
⋅
i
r
S_{k}^{*}=\sum_{\{i_{1},i_{2},...,i_{r}\}\in S_{k}}\frac{1}{i_{1}\cdot i_{2}\cdot...\cdot i_{r}}
S
k
∗
=
{
i
1
,
i
2
,
...
,
i
r
}
∈
S
k
∑
i
1
⋅
i
2
⋅
...
⋅
i
r
1
in fact, this problem was taken from an austrian-polish
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