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Putnam
1940 Putnam
A6
A6
Part of
1940 Putnam
Problems
(1)
Putnam 1940 A6
Source: Putnam 1940
2/22/2022
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial of degree
n
n
n
such that
f
(
x
)
p
f(x)^{p}
f
(
x
)
p
is divisible by
f
′
(
x
)
q
f'(x)^{q}
f
′
(
x
)
q
for some positive integers
p
,
q
p,q
p
,
q
. Prove that
f
(
x
)
f(x)
f
(
x
)
is divisible by
f
′
(
x
)
f'(x)
f
′
(
x
)
and that
f
(
x
)
f(x)
f
(
x
)
has a single root of multiplicity
n
n
n
.
Putnam
algebra
polynomial