MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2016
P2
Weird strict inequality
Weird strict inequality
Source: Russian TST 2016, Day 10 P2 (Group NG)
April 19, 2023
algebra
inequalities
Problem Statement
Let
x
,
y
,
z
x,y,z{}
x
,
y
,
z
be positive real numbers. Prove that
(
x
y
+
y
z
+
z
x
)
(
1
x
2
+
y
2
+
1
y
2
+
z
2
+
1
z
2
+
x
2
)
>
5
2
.
(xy+yz+zx)\left(\frac{1}{x^2+y^2}+\frac{1}{y^2+z^2}+\frac{1}{z^2+x^2}\right)>\frac{5}{2}.
(
x
y
+
yz
+
z
x
)
(
x
2
+
y
2
1
+
y
2
+
z
2
1
+
z
2
+
x
2
1
)
>
2
5
.
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