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Putnam
1957 Putnam
A4
A4
Part of
1957 Putnam
Problems
(1)
Roots of polynomial covered by round disk
Source:
6/4/2019
Let
P
(
z
)
P(z)
P
(
z
)
be a polynomial with real coefficients whose roots are covered by a disk of radius R. Prove that for any real number
k
k
k
, the roots of the polynomial
n
P
(
z
)
−
k
P
′
(
z
)
nP(z)-kP'(z)
n
P
(
z
)
−
k
P
′
(
z
)
can be covered by a disk of radius
R
+
∣
k
∣
R+|k|
R
+
∣
k
∣
, where
n
n
n
is the degree of
P
(
z
)
P(z)
P
(
z
)
, and
P
′
(
z
)
P'(z)
P
′
(
z
)
is the derivative of
P
(
z
)
P(z)
P
(
z
)
. can anyone help me? It would also be extremely helpful if anyone could tell me where they've seen this type of problems.............Has it appeared in any mathematics competitions? Or are there any similar questions for me to attempt? Thanks in advance!
algebra
polynomial