MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
Macedonia National Olympiad
2017 Macedonia National Olympiad
Problem 3
Macedonia National Olympiad 2017 Problem 3 Inequality
Macedonia National Olympiad 2017 Problem 3 Inequality
Source: Macedonia National Olympiad 2017, Problem 3
April 8, 2017
inequalities
Problem Statement
Let
x
,
y
,
z
∈
R
x,y,z \in \mathbb{R}
x
,
y
,
z
∈
R
such that
x
y
z
=
1
xyz = 1
x
yz
=
1
. Prove that:
(
x
4
+
z
2
y
2
)
(
y
4
+
x
2
z
2
)
(
z
4
+
y
2
x
2
)
≥
(
x
2
y
+
1
)
(
y
2
z
+
1
)
(
z
2
x
+
1
)
.
\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).
(
x
4
+
y
2
z
2
)
(
y
4
+
z
2
x
2
)
(
z
4
+
x
2
y
2
)
≥
(
y
x
2
+
1
)
(
z
y
2
+
1
)
(
x
z
2
+
1
)
.
Back to Problems
View on AoPS