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Source:
January 6, 2022
Summation
inequalities
binomial coefficients
algebra
Problem Statement
Prove that for all positive integers
m
m
m
and
n
n
n
,
1
m
⋅
(
2
n
0
)
−
1
m
+
1
⋅
(
2
n
1
)
+
1
m
+
2
⋅
(
2
n
2
)
−
…
+
1
m
+
2
n
⋅
(
2
n
n
2
)
>
0
\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0
m
1
⋅
(
0
2
n
)
−
m
+
1
1
⋅
(
1
2
n
)
+
m
+
2
1
⋅
(
2
2
n
)
−
…
+
m
+
2
n
1
⋅
(
n
2
2
n
)
>
0
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