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C4
C4
Part of
2014 Indonesia MO Shortlist
Problems
(1)
Binomial coefficients everywhere
Source: Indonesian Mathematical Olympiad 2014 Day 2 Problem 7
9/4/2014
Suppose that
k
,
m
,
n
k,m,n
k
,
m
,
n
are positive integers with
k
≤
n
k \le n
k
≤
n
. Prove that:
∑
r
=
0
m
k
(
m
r
)
(
n
k
)
(
r
+
k
)
(
m
+
n
r
+
k
)
=
1
\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1
r
=
0
∑
m
(
r
+
k
)
(
r
+
k
m
+
n
)
k
(
r
m
)
(
k
n
)
=
1
algebra
polynomial