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South Africa National Olympiad
2001 South africa National Olympiad
2
2
Part of
2001 South africa National Olympiad
Problems
(1)
Complicated equations :)
Source: South Africa 2001
10/1/2005
Find all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers that satisfy
x
(
1
−
y
2
)
(
1
−
z
2
)
+
y
(
1
−
z
2
)
(
1
−
x
2
)
+
z
(
1
−
x
2
)
(
1
−
y
2
)
=
4
x
y
z
=
4
(
x
+
y
+
z
)
.
\begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) \\ & = 4xyz \\ & = 4(x + y + z). \end{aligned}
x
(
1
−
y
2
)
(
1
−
z
2
)
+
y
(
1
−
z
2
)
(
1
−
x
2
)
+
z
(
1
−
x
2
)
(
1
−
y
2
)
=
4
x
yz
=
4
(
x
+
y
+
z
)
.
trigonometry
algebra unsolved
algebra