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Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2006 Vietnam National Olympiad
1
Vmo 2006 a1
Vmo 2006 a1
Source:
February 28, 2006
logarithms
function
algebra
system of equations
algebra unsolved
Problem Statement
Solve the following system of equations in real numbers:
{
x
2
−
2
x
+
6
⋅
log
3
(
6
−
y
)
=
x
y
2
−
2
y
+
6
⋅
log
3
(
6
−
z
)
=
y
z
2
−
2
z
+
6
⋅
log
3
(
6
−
x
)
=
z
.
\begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}.
⎩
⎨
⎧
x
2
−
2
x
+
6
⋅
lo
g
3
(
6
−
y
)
=
x
y
2
−
2
y
+
6
⋅
lo
g
3
(
6
−
z
)
=
y
z
2
−
2
z
+
6
⋅
lo
g
3
(
6
−
x
)
=
z
.
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