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Putnam
1951 Putnam
B6
Putnam 1951 B6
Putnam 1951 B6
Source:
May 25, 2022
Putnam
Problem Statement
Assuming that all of the roots of the cubic equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3 + ax^2 +bx + c = 0
x
3
+
a
x
2
+
b
x
+
c
=
0
are real, show that the difference between the greatest and the least roots is not less than
(
a
2
−
3
b
)
1
/
2
(a^2 - 3b)^{1/2}
(
a
2
−
3
b
)
1/2
or greater than
2
(
a
2
−
3
b
)
1
/
2
/
3
1
/
2
.
2 (a^2 - 3b)^{1/2} / 3^{1/2}.
2
(
a
2
−
3
b
)
1/2
/
3
1/2
.
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