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Tuymaada Olympiad
2013 Tuymaada Olympiad
4
4
Part of
2013 Tuymaada Olympiad
Problems
(1)
Three positive variables with product 1
Source: Tuymaada 2013, Day 1, Problem 4 Seniors
7/24/2013
Prove that if
x
x
x
,
y
y
y
,
z
z
z
are positive real numbers and
x
y
z
=
1
xyz = 1
x
yz
=
1
then
x
3
x
2
+
y
+
y
3
y
2
+
z
+
z
3
z
2
+
x
≥
3
2
.
\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.
x
2
+
y
x
3
+
y
2
+
z
y
3
+
z
2
+
x
z
3
≥
2
3
.
A. Golovanov
inequalities
Tuymaada
three variable inequality