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Azerbaijan Junior National Olympiad
2024 Azerbaijan National Mathematical Olympiad
2
finding triples using inequalities
finding triples using inequalities
Source: Azerbaijan NMO 2024. Junior P2
July 8, 2024
algebra
Inequality
inequalities
AZE JUNIOR NATIONAL MO
Problem Statement
Find all the real number triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
satisfying the following system of inequalities under the condition
0
<
x
,
y
,
z
<
2
0 < x, y, z < \sqrt{2}
0
<
x
,
y
,
z
<
2
:
y
4
−
x
2
y
2
≥
2
x
z
y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}
y
4
−
x
2
y
2
≥
x
z
2
x
4
−
x
2
z
2
≥
2
y
z
x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}
x
4
−
x
2
z
2
≥
yz
2
z
4
−
y
2
z
2
≥
2
x
y
z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}
z
4
−
y
2
z
2
≥
x
y
2
.
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