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Saint Petersburg Mathematical Olympiad
2012 Saint Petersburg Mathematical Olympiad
1
Common roots
Common roots
Source: St Petersburg Olympiad 2012, Grade 11, P1
September 29, 2017
algebra
Problem Statement
a
,
b
,
c
a,b,c
a
,
b
,
c
are reals, such that every pair of equations of
x
3
−
a
x
2
+
b
=
0
,
x
3
−
b
x
2
+
c
=
0
,
x
3
−
c
x
2
+
a
=
0
x^3-ax^2+b=0,x^3-bx^2+c=0,x^3-cx^2+a=0
x
3
−
a
x
2
+
b
=
0
,
x
3
−
b
x
2
+
c
=
0
,
x
3
−
c
x
2
+
a
=
0
has common root. Prove
a
=
b
=
c
a=b=c
a
=
b
=
c
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