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IMO Longlists
1974 IMO Longlists
32
Familiar inequality.. [ILL 1971]
Familiar inequality.. [ILL 1971]
Source:
January 2, 2011
inequalities
inequalities proposed
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots ,a_n
a
1
,
a
2
,
…
,
a
n
be
n
n
n
real numbers such that
0
<
a
≤
a
k
≤
b
0<a\le a_k\le b
0
<
a
≤
a
k
≤
b
for
k
=
1
,
2
,
…
,
n
k=1,2,\ldots ,n
k
=
1
,
2
,
…
,
n
. If
m
1
=
1
n
(
a
1
+
a
2
+
⋯
+
a
n
)
m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)
m
1
=
n
1
(
a
1
+
a
2
+
⋯
+
a
n
)
and
m
2
=
1
n
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)
m
2
=
n
1
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
, prove that
m
2
≤
(
a
+
b
)
2
4
a
b
m
1
2
m_2\le\frac{(a+b)^2}{4ab}m_1^2
m
2
≤
4
ab
(
a
+
b
)
2
m
1
2
and find a necessary and sufficient condition for equality.
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