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Putnam
1997 Putnam
5
5
Part of
1997 Putnam
Problems
(2)
Putnam 1997 A5
Source:
5/30/2014
Let
N
k
N_k
N
k
denote number of ordered
n
n
n
-tuples of positive integers
(
a
1
,
a
2
,
⋯
,
a
k
)
(a_1,a_2, \cdots ,a_k)
(
a
1
,
a
2
,
⋯
,
a
k
)
such that
1
a
1
+
1
a
2
+
…
+
1
a
k
=
1
\frac{1}{a_1}+\frac{1}{a_2}+\ldots +\frac{1}{a_k}=1
a
1
1
+
a
2
1
+
…
+
a
k
1
=
1
Determine
N
10
N_{10}
N
10
is odd or even.
Putnam
college contests
Putnam 1997 B5
Source:
5/30/2014
Let us define a sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\ge 1}
{
a
n
}
n
≥
1
. Define as follows:
a
1
=
2
and
a
n
+
1
=
2
a
n
for
n
≥
1
a_1=2\text{ and }a_{n+1}=2^{a_n}\text{ for }n\ge 1
a
1
=
2
and
a
n
+
1
=
2
a
n
for
n
≥
1
Show this :
a
n
≡
a
n
−
1
(
m
o
d
n
)
a_{n}\equiv a_{n-1}\pmod n
a
n
≡
a
n
−
1
(
mod
n
)
Putnam
modular arithmetic
college contests