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Putnam
1941 Putnam
A4
A4
Part of
1941 Putnam
Problems
(1)
Putnam 1941 A4
Source: Putnam 1941
2/23/2022
Let the roots
a
,
b
,
c
a,b,c
a
,
b
,
c
of
f
(
x
)
=
x
3
+
p
x
2
+
q
x
+
r
f(x)=x^3 +p x^2 + qx+r
f
(
x
)
=
x
3
+
p
x
2
+
q
x
+
r
be real, and let
a
≤
b
≤
c
a\leq b\leq c
a
≤
b
≤
c
. Prove that
f
′
(
x
)
f'(x)
f
′
(
x
)
has a root in the interval
[
b
+
c
2
,
b
+
2
c
3
]
\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]
[
2
b
+
c
,
3
b
+
2
c
]
. What will be the form of
f
(
x
)
f(x)
f
(
x
)
if the root in question falls at either end of the interval?
Putnam
polynomial
roots