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Putnam
1997 Putnam
3
3
Part of
1997 Putnam
Problems
(2)
Putnam 1997 A3
Source:
5/30/2014
Evaluate the following :
∫
0
∞
(
x
−
x
3
2
+
x
5
2
⋅
4
−
x
7
2
⋅
4
⋅
6
+
⋯
)
(
1
+
x
2
2
2
+
x
4
2
2
⋅
4
2
+
x
6
2
2
⋅
4
2
⋅
6
2
+
⋯
)
d
x
\int_{0}^{\infty}\left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots \right)\;\left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2\cdot 6^2}+\cdots \right)\,\mathrm{d}x
∫
0
∞
(
x
−
2
x
3
+
2
⋅
4
x
5
−
2
⋅
4
⋅
6
x
7
+
⋯
)
(
1
+
2
2
x
2
+
2
2
⋅
4
2
x
4
+
2
2
⋅
4
2
⋅
6
2
x
6
+
⋯
)
d
x
Putnam
integration
calculus
induction
college contests
Putnam 1997 B3
Source:
5/30/2014
For each positive integer
n
n
n
write the sum
∑
i
=
n
1
i
=
p
n
q
n
\sum_{i=}^{n}\frac{1}{i}=\frac{p_n}{q_n}
∑
i
=
n
i
1
=
q
n
p
n
with
gcd
(
p
n
,
q
n
)
=
1
\text{gcd}(p_n,q_n)=1
gcd
(
p
n
,
q
n
)
=
1
. Find all such
n
n
n
such that
5
∤
q
n
5\nmid q_n
5
∤
q
n
.
Putnam
number theory
greatest common divisor
college contests