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2015 JBMO Shortlist
A5
Simple inequality
Simple inequality
Source: JBMO Shortlist 2015 , A5 (FYROM)
May 20, 2016
inequalities
JBMO
China
BPSQ
JBMO Shortlist
High school olympiad
Problem Statement
The positive real
x
,
y
,
z
x, y, z
x
,
y
,
z
are such that
x
2
+
y
2
+
z
2
=
3
x^2+y^2+z^2 = 3
x
2
+
y
2
+
z
2
=
3
. Prove that
x
2
+
y
z
x
2
+
y
z
+
1
+
y
2
+
z
x
y
2
+
z
x
+
1
+
z
2
+
x
y
z
2
+
x
y
+
1
≤
2
\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2
x
2
+
yz
+
1
x
2
+
yz
+
y
2
+
z
x
+
1
y
2
+
z
x
+
z
2
+
x
y
+
1
z
2
+
x
y
≤
2
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