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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2000 China Team Selection Test
2
Faculty fraction sum
Faculty fraction sum
Source: China TST 2000, problem 2
May 22, 2005
induction
algebra
polynomial
factorial
combinatorics
Finite Differences
Problem Statement
Given positive integers
k
,
m
,
n
k, m, n
k
,
m
,
n
such that
1
≤
k
≤
m
≤
n
1 \leq k \leq m \leq n
1
≤
k
≤
m
≤
n
. Evaluate
∑
i
=
0
n
(
−
1
)
i
n
+
k
+
i
⋅
(
m
+
n
+
i
)
!
i
!
(
n
−
i
)
!
(
m
+
i
)
!
.
\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.
i
=
0
∑
n
n
+
k
+
i
(
−
1
)
i
⋅
i
!
(
n
−
i
)!
(
m
+
i
)!
(
m
+
n
+
i
)!
.
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