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Balkan MO
1984 Balkan MO
4
4
Part of
1984 Balkan MO
Problems
(1)
system of equations
Source: bmo 1984
4/23/2007
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Find all real solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of the system:
a
x
+
b
y
=
(
x
−
y
)
2
b
y
+
c
z
=
(
y
−
z
)
2
c
z
+
a
x
=
(
z
−
x
)
2
ax+by=(x-y)^{2} \\ by+cz=(y-z)^{2} \\ cz+ax=(z-x)^{2}
a
x
+
b
y
=
(
x
−
y
)
2
b
y
+
cz
=
(
y
−
z
)
2
cz
+
a
x
=
(
z
−
x
)
2
algebra proposed
algebra