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National and Regional Contests
PEN Problems
PEN S Problems
9
S 9
S 9
Source:
May 25, 2007
Miscellaneous Problems
Problem Statement
Suppose that
∏
n
=
1
1996
(
1
+
n
x
3
n
)
=
1
+
a
1
x
k
1
+
a
2
x
k
2
+
⋯
+
a
m
x
k
m
\prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\cdots+a_{m}x^{k_{m}}
n
=
1
∏
1996
(
1
+
n
x
3
n
)
=
1
+
a
1
x
k
1
+
a
2
x
k
2
+
⋯
+
a
m
x
k
m
where
a
1
a_{1}
a
1
,
a
2
a_{2}
a
2
,...,
a
m
a_{m}
a
m
are nonzero and
k
1
<
k
2
<
⋯
<
k
m
k_{1}< k_{2}< \cdots < k_{m}
k
1
<
k
2
<
⋯
<
k
m
. Find
a
1996
a_{1996}
a
1996
.
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