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Putnam
1953 Putnam
B2
B2
Part of
1953 Putnam
Problems
(1)
Putnam 1953 B2
Source: Putnam 1953
7/16/2022
Let
a
0
,
a
1
,
…
,
a
n
a_0 ,a_1 , \ldots, a_n
a
0
,
a
1
,
…
,
a
n
be real numbers and let
f
(
x
)
=
a
n
x
n
+
…
+
a
1
x
+
a
0
.
f(x) =a_n x^n +\ldots +a_1 x +a_0.
f
(
x
)
=
a
n
x
n
+
…
+
a
1
x
+
a
0
.
Suppose that
f
(
i
)
f(i)
f
(
i
)
is an integer for all
i
.
i.
i
.
Prove that
n
!
⋅
a
k
n! \cdot a_k
n
!
⋅
a
k
is an integer for each
k
.
k.
k
.
Putnam
Integer
polynomial