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Mediterranean Mathematics Olympiad
2009 Mediterranean Mathematics Olympiad
4
3-variable Inequality
3-variable Inequality
Source: 2009 MMO Problem #4
September 12, 2011
inequalities
inequalities unsolved
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers. Prove that
∑
c
y
c
l
i
c
x
y
x
y
+
x
2
+
y
2
≤
∑
c
y
c
l
i
c
x
2
x
+
z
\sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z}
cyc
l
i
c
∑
x
y
+
x
2
+
y
2
x
y
≤
cyc
l
i
c
∑
2
x
+
z
x
(Proposed by Šefket Arslanagić, Bosnia and Herzegovina)
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