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Mediterranean Mathematics Olympiad
1999 Mediterranean Mathematics Olympiad
3
Strange Inequality
Strange Inequality
Source: Mediterranean MO 1999
October 11, 2012
inequalities
inequalities unsolved
Problem Statement
Let
a
,
b
,
c
≠
0
a,b,c\not= 0
a
,
b
,
c
=
0
and
x
,
y
,
z
∈
R
+
x,y,z\in\mathbb{R}^+
x
,
y
,
z
∈
R
+
such that
x
+
y
+
z
=
3
x+y+z=3
x
+
y
+
z
=
3
. Prove that
3
2
1
a
2
+
1
b
2
+
1
c
2
≥
x
1
+
a
2
+
y
1
+
b
2
+
z
1
+
c
2
\frac{3}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}
2
3
a
2
1
+
b
2
1
+
c
2
1
≥
1
+
a
2
x
+
1
+
b
2
y
+
1
+
c
2
z
[color=#FF0000]Mod: before the edit, it was
3
2
(
1
a
2
+
1
b
2
+
1
c
2
)
≥
x
1
+
a
2
+
y
1
+
b
2
+
z
1
+
c
2
\frac{3}{2}\left (\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right )\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}
2
3
(
a
2
1
+
b
2
1
+
c
2
1
)
≥
1
+
a
2
x
+
1
+
b
2
y
+
1
+
c
2
z
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