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1992 IMO Longlists
57
Nice inequality with A,G,H - IMO LongList 1992
Nice inequality with A,G,H - IMO LongList 1992
Source:
September 2, 2010
function
three variable inequality
Inequality
algebra
IMO Shortlist
IMO Longlist
Problem Statement
For positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
define
A
=
(
a
+
b
+
c
)
3
A = \frac{(a + b + c)}{3}
A
=
3
(
a
+
b
+
c
)
,
G
=
a
b
c
3
G = \sqrt[3]{abc}
G
=
3
ab
c
,
H
=
3
(
a
−
1
+
b
−
1
+
c
−
1
)
.
H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.
H
=
(
a
−
1
+
b
−
1
+
c
−
1
)
3
.
Prove that
(
A
G
)
3
≥
1
4
+
3
4
⋅
A
H
.
\left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.
(
G
A
)
3
≥
4
1
+
4
3
⋅
H
A
.
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