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Ireland National Math Olympiad
2015 Irish Math Olympiad
10
10
Part of
2015 Irish Math Olympiad
Problems
(1)
binomial inequality, with pairs of nonnegative integers
Source: Irmo 2015 p2 q10
9/16/2018
Prove that, for all pairs of nonnegative integers,
j
,
n
j,n
j
,
n
,
∑
K
=
0
n
k
j
(
n
k
)
≥
2
n
−
j
n
j
\sum_{K=0}^{n}k^j\binom n k \ge 2^{n-j} n^j
K
=
0
∑
n
k
j
(
k
n
)
≥
2
n
−
j
n
j
Binomial
inequalities
Binomial summation
Inequality