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National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2009 Serbia National Math Olympiad
5
5
Part of
2009 Serbia National Math Olympiad
Problems
(1)
Serbian MO 2009
Source:
1/24/2016
Let
x
x
x
,
y
y
y
,
z
z
z
be arbitrary positive numbers such that
x
y
+
y
z
+
z
x
=
x
+
y
+
z
xy+yz+zx=x+y+z
x
y
+
yz
+
z
x
=
x
+
y
+
z
. Prove that
1
x
2
+
y
+
1
+
1
y
2
+
z
+
1
+
1
z
2
+
x
+
1
≤
1
\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1
x
2
+
y
+
1
1
+
y
2
+
z
+
1
1
+
z
2
+
x
+
1
1
≤
1
. When does equality occur? Proposed by Marko Radovanovic
inequalities
three variable inequality
Serbia