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2023 Harvard-MIT Mathematics Tournament
8
HMMT Feb 2023 team p8
HMMT Feb 2023 team p8
Source:
February 20, 2023
Problem Statement
Find, with proof, all nonconstant polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that, for all nonzero real numbers
z
z
z
with
P
(
z
)
≠
0
P(z)\neq 0
P
(
z
)
=
0
and
P
(
1
z
)
≠
0
P(\frac{1}{z}) \neq 0
P
(
z
1
)
=
0
we have
1
P
(
z
)
+
1
P
(
1
z
)
=
z
+
1
z
.
\frac{1}{P(z)}+\frac{1}{P(\frac{1} {z})}=z+\frac{1}{z}.
P
(
z
)
1
+
P
(
z
1
)
1
=
z
+
z
1
.
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