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Putnam
2005 Putnam
A3
A3
Part of
2005 Putnam
Problems
(1)
Putnam 2005 A3
Source:
12/5/2005
Let
p
(
z
)
p(z)
p
(
z
)
be a polynomial of degree
n
,
n,
n
,
all of whose zeros have absolute value
1
1
1
in the complex plane. Put
g
(
z
)
=
p
(
z
)
z
n
/
2
.
g(z)=\frac{p(z)}{z^{n/2}}.
g
(
z
)
=
z
n
/2
p
(
z
)
.
Show that all zeros of
g
′
(
z
)
=
0
g'(z)=0
g
′
(
z
)
=
0
have absolute value
1.
1.
1.
Putnam
algebra
polynomial
Stanford
college
function
geometry