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1988 Bulgaria National Olympiad
Problem 1
parameter, quartic has four roots
parameter, quartic has four roots
Source: Bulgaria 1988 P1
June 15, 2021
algebra
Problem Statement
Find all real parameters
q
q
q
for which there is a
p
∈
[
0
,
1
]
p\in[0,1]
p
∈
[
0
,
1
]
such that the equation
x
4
+
2
p
x
3
+
(
2
p
2
−
p
)
x
2
+
(
p
−
1
)
p
2
x
+
q
=
0
x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0
x
4
+
2
p
x
3
+
(
2
p
2
−
p
)
x
2
+
(
p
−
1
)
p
2
x
+
q
=
0
has four real roots.
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