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20
2007 Guts #20: Roots of an Equation
2007 Guts #20: Roots of an Equation
Source:
June 22, 2012
algebra
polynomial
Vieta
Problem Statement
For
a
a
a
a positive real number, let
x
1
x_1
x
1
,
x
2
x_2
x
2
,
x
3
x_3
x
3
be the roots of the equation
x
3
−
a
x
2
+
a
x
−
a
=
0
x^3-ax^2+ax-a=0
x
3
−
a
x
2
+
a
x
−
a
=
0
. Determine the smallest possible value of
x
1
3
+
x
2
3
+
x
3
3
−
3
x
1
x
2
x
3
x_1^3+x_2^3+x_3^3-3x_1x_2x_3
x
1
3
+
x
2
3
+
x
3
3
−
3
x
1
x
2
x
3
.
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