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Baltic Way
2022 Baltic Way
4
Inequality with xy+yz+zx=1
Inequality with xy+yz+zx=1
Source: Baltic Way 2022, Problem 4
November 12, 2022
inequalities
Problem Statement
The positive real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfy
x
y
+
y
z
+
z
x
=
1
xy+yz+zx=1
x
y
+
yz
+
z
x
=
1
. Prove that:
2
(
x
2
+
y
2
+
z
2
)
+
4
3
(
1
x
2
+
1
+
1
y
2
+
1
+
1
z
2
+
1
)
≥
5
2(x^2+y^2+z^2)+\frac{4}{3}\bigg (\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}\bigg) \ge 5
2
(
x
2
+
y
2
+
z
2
)
+
3
4
(
x
2
+
1
1
+
y
2
+
1
1
+
z
2
+
1
1
)
≥
5
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