MathDB
Problems
Contests
International Contests
IMO Longlists
1979 IMO Longlists
63
63
Part of
1979 IMO Longlists
Problems
(1)
Classic polygon inequality
Source: ILL 1979 - Problem 63.
6/5/2011
Let the sequence
{
a
i
}
\{a_i\}
{
a
i
}
of
n
n
n
positive reals denote the lengths of the sides of an arbitrary
n
n
n
-gon. Let
s
=
∑
i
=
1
n
a
i
s=\sum_{i=1}^{n}{a_i}
s
=
∑
i
=
1
n
a
i
. Prove that
2
≥
∑
i
=
1
n
a
i
s
−
a
i
≥
n
n
−
1
2\ge \sum_{i=1}^{n}{\frac{a_i}{s-a_i}}\ge \frac{n}{n-1}
2
≥
∑
i
=
1
n
s
−
a
i
a
i
≥
n
−
1
n
.
inequalities
triangle inequality
inequalities proposed