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Putnam
1950 Putnam
A6
A6
Part of
1950 Putnam
Problems
(1)
Putnam 1950 A6
Source:
5/24/2022
Each coefficient
a
n
a_n
a
n
of the power series
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
+
⋯
=
f
(
x
)
a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots = f(x)
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
+
⋯
=
f
(
x
)
has either the value of
1
1
1
or the value
0.
0.
0.
Prove the easier of the two assertions:(i) If
f
(
0.5
)
f(0.5)
f
(
0.5
)
is a rational number,
f
(
x
)
f(x)
f
(
x
)
is a rational function.(ii) If
f
(
0.5
)
f(0.5)
f
(
0.5
)
is not a rational number,
f
(
x
)
f(x)
f
(
x
)
is not a rational function.
Putnam