MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2008 Greece National Olympiad
4
4
Part of
2008 Greece National Olympiad
Problems
(1)
Inequality for n integers
Source: Greek National Mathematical Olympiad 2008 - P4
11/17/2011
If
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots , a_n
a
1
,
a
2
,
…
,
a
n
are positive integers and
k
=
max
{
a
1
,
…
,
a
n
}
k = \max\{a_1, \ldots, a_n\}
k
=
max
{
a
1
,
…
,
a
n
}
,
t
=
min
{
a
1
,
…
,
a
n
}
t = \min\{a_1,\ldots, a_n\}
t
=
min
{
a
1
,
…
,
a
n
}
, prove the inequality
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
a
1
+
a
2
+
⋯
+
a
n
)
k
n
t
≥
a
1
a
2
⋯
a
n
.
\left(\frac{a_1^2+a_2^2+\cdots+a_n^2}{a_1+a_2+\cdots+a_n}\right)^{\frac{kn}{t}} \geq a_1a_2\cdots a_n.
(
a
1
+
a
2
+
⋯
+
a
n
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
t
kn
≥
a
1
a
2
⋯
a
n
.
When does equality hold?
inequalities
inequalities unsolved