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Putnam
2018 Putnam
B2
Putnam 2018 B2
Putnam 2018 B2
Source:
December 2, 2018
Putnam
Putnam 2018
Problem Statement
Let
n
n
n
be a positive integer, and let
f
n
(
z
)
=
n
+
(
n
−
1
)
z
+
(
n
−
2
)
z
2
+
⋯
+
z
n
−
1
f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}
f
n
(
z
)
=
n
+
(
n
−
1
)
z
+
(
n
−
2
)
z
2
+
⋯
+
z
n
−
1
. Prove that
f
n
f_n
f
n
has no roots in the closed unit disk
{
z
∈
C
:
∣
z
∣
≤
1
}
\{z \in \mathbb{C}: |z| \le 1\}
{
z
∈
C
:
∣
z
∣
≤
1
}
.
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