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Romania Team Selection Test
1990 Romania Team Selection Test
2
binomial sums equality
binomial sums equality
Source: Romania IMO TST 1990 p2
February 19, 2020
algebra
Binomial
Sum
combinatorics
Problem Statement
Prove the following equality for all positive integers
m
,
n
m,n
m
,
n
:
∑
k
=
0
n
(
m
+
k
k
)
2
n
−
k
+
∑
k
=
0
m
(
n
+
k
k
)
2
m
−
k
=
2
m
+
n
+
1
\sum_{k=0}^{n} {m+k \choose k} 2^{n-k} +\sum_{k=0}^m {n+k \choose k}2^{m-k}= 2^{m+n+1}
k
=
0
∑
n
(
k
m
+
k
)
2
n
−
k
+
k
=
0
∑
m
(
k
n
+
k
)
2
m
−
k
=
2
m
+
n
+
1
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