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Saint Petersburg Mathematical Olympiad
2012 Saint Petersburg Mathematical Olympiad
1
system of cubic
system of cubic
Source: St Petersburg Olympiad 2012, Grade 10, P1
September 29, 2017
algebra
Problem Statement
{
x
3
−
a
x
2
+
b
3
=
0
x
3
−
b
x
2
+
c
3
=
0
x
3
−
c
x
2
+
a
3
=
0
\begin{cases} x^3-ax^2+b^3=0 \\x^3-bx^2+c^3=0 \\ x^3-cx^2+a^3=0 \end{cases}
⎩
⎨
⎧
x
3
−
a
x
2
+
b
3
=
0
x
3
−
b
x
2
+
c
3
=
0
x
3
−
c
x
2
+
a
3
=
0
Prove that system hasn`t solutions if
a
,
b
,
c
a,b,c
a
,
b
,
c
are different.
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