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Putnam
2023 Putnam
A2
A2
Part of
2023 Putnam
Problems
(1)
2023 Putnam A2
Source:
12/3/2023
Let
n
n
n
be an even positive integer. Let
p
p
p
be a monic, real polynomial of degree
2
n
2 n
2
n
; that is to say,
p
(
x
)
=
p(x)=
p
(
x
)
=
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
⋯
+
a
1
x
+
a
0
x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
⋯
+
a
1
x
+
a
0
for some real coefficients
a
0
,
…
,
a
2
n
−
1
a_0, \ldots, a_{2 n-1}
a
0
,
…
,
a
2
n
−
1
. Suppose that
p
(
1
/
k
)
=
k
2
p(1 / k)=k^2
p
(
1/
k
)
=
k
2
for all integers
k
k
k
such that
1
≤
∣
k
∣
≤
n
1 \leq|k| \leq n
1
≤
∣
k
∣
≤
n
. Find all other real numbers
x
x
x
for which
p
(
1
/
x
)
=
x
2
p(1 / x)=x^2
p
(
1/
x
)
=
x
2
.
Putnam
Putnam 2023