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Iran Team Selection Test
2019 Iran Team Selection Test
6
Iran inequality
Iran inequality
Source: Iranian TST 2019, third exam day 2, problem 6
April 15, 2019
inequalities
Problem Statement
x
,
y
x,y
x
,
y
and
z
z
z
are real numbers such that
x
+
y
+
z
=
x
y
+
y
z
+
z
x
x+y+z=xy+yz+zx
x
+
y
+
z
=
x
y
+
yz
+
z
x
. Prove that
x
x
4
+
x
2
+
1
+
y
y
4
+
y
2
+
1
+
z
z
4
+
z
2
+
1
≥
−
1
3
.
\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.
x
4
+
x
2
+
1
x
+
y
4
+
y
2
+
1
y
+
z
4
+
z
2
+
1
z
≥
3
−
1
.
Proposed by Navid Safaei
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