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1970 IMO Longlists
24
Easy Factorial Inequality
Easy Factorial Inequality
Source: ILL 1970 - Problem 24.
May 24, 2011
factorial
inequalities
inequalities proposed
Problem Statement
Let
{
n
,
p
}
∈
N
∪
{
0
}
\{n,p\}\in\mathbb{N}\cup \{0\}
{
n
,
p
}
∈
N
∪
{
0
}
such that
2
p
≤
n
2p\le n
2
p
≤
n
. Prove that
(
n
−
p
)
!
p
!
≤
(
n
+
1
2
)
n
−
2
p
\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}
p
!
(
n
−
p
)!
≤
(
2
n
+
1
)
n
−
2
p
. Determine all conditions under which equality holds.
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