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IMO Shortlist
1975 IMO Shortlist
7
For every x,y such that x+y=1 prove the sum is equal to 1
For every x,y such that x+y=1 prove the sum is equal to 1
Source:
September 21, 2010
algebra
Summation
binomial coefficients
Binomial coefficient identity
combinatorics
IMO Shortlist
Problem Statement
Prove that from
x
+
y
=
1
(
x
,
y
∈
R
)
x + y = 1 \ (x, y \in \mathbb R)
x
+
y
=
1
(
x
,
y
∈
R
)
it follows that
x
m
+
1
∑
j
=
0
n
(
m
+
j
j
)
y
j
+
y
n
+
1
∑
i
=
0
m
(
n
+
i
i
)
x
i
=
1
(
m
,
n
=
0
,
1
,
2
,
…
)
.
x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).
x
m
+
1
j
=
0
∑
n
(
j
m
+
j
)
y
j
+
y
n
+
1
i
=
0
∑
m
(
i
n
+
i
)
x
i
=
1
(
m
,
n
=
0
,
1
,
2
,
…
)
.
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