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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2020 Vietnam National Olympiad
5
5
Part of
2020 Vietnam National Olympiad
Problems
(1)
VMO-D2-P5
Source: VMO-2020 D2-P5
12/28/2019
Let a system of equations:
{
x
−
a
y
=
y
z
y
−
a
z
=
z
x
z
−
a
x
=
x
y
\left\{\begin{matrix}x-ay=yz\\y-az=zx\\z-ax=xy\end{matrix}\right.
⎩
⎨
⎧
x
−
a
y
=
yz
y
−
a
z
=
z
x
z
−
a
x
=
x
y
a)Find (x,y,z) if a=0 b)Prove that: the system have 5 distinct roots
∀
\forall
∀
a>1,a
∈
R
.
\in\mathbb{R}.
∈
R
.
system of equations
algebra