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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1979 Vietnam National Olympiad
2
2
Part of
1979 Vietnam National Olympiad
Problems
(1)
x^3 + ax^2 + bx + c , x^3 +α^3 x^2 + β^3 x + γ^3, 3 real roots each, related
Source: Vietnam MO 1979 P2
7/28/2018
Find all real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
x
3
+
a
x
2
+
b
x
+
c
x^3 + ax^2 + bx + c
x
3
+
a
x
2
+
b
x
+
c
has three real roots
α
,
β
,
γ
\alpha, \beta,\gamma
α
,
β
,
γ
(not necessarily all distinct) and the equation
x
3
+
α
3
x
2
+
β
3
x
+
γ
3
x^3 + \alpha^3 x^2 + \beta^3 x + \gamma^3
x
3
+
α
3
x
2
+
β
3
x
+
γ
3
has roots
α
3
,
β
3
,
γ
3
\alpha^3, \beta^3,\gamma^3
α
3
,
β
3
,
γ
3
.
algebra
Polynomials