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Putnam
2002 Putnam
5
Putnam 2002 A5
Putnam 2002 A5
Source:
March 12, 2012
Putnam
induction
ratio
number theory
relatively prime
college contests
Problem Statement
Define a sequence by
a
0
=
1
a_0=1
a
0
=
1
, together with the rules
a
2
n
+
1
=
a
n
a_{2n+1}=a_n
a
2
n
+
1
=
a
n
and
a
2
n
+
2
=
a
n
+
a
n
+
1
a_{2n+2}=a_n+a_{n+1}
a
2
n
+
2
=
a
n
+
a
n
+
1
for each integer
n
≥
0
n\ge0
n
≥
0
. Prove that every positive rational number appears in the set
{
a
n
−
1
a
n
:
n
≥
1
}
=
{
1
1
,
1
2
,
2
1
,
1
3
,
3
2
,
⋯
}
\left\{ \tfrac {a_{n-1}}{a_n}: n \ge 1 \right\} = \left\{ \tfrac {1}{1}, \tfrac {1}{2}, \tfrac {2}{1}, \tfrac {1}{3}, \tfrac {3}{2}, \cdots \right\}
{
a
n
a
n
−
1
:
n
≥
1
}
=
{
1
1
,
2
1
,
1
2
,
3
1
,
2
3
,
⋯
}
.
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