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Putnam
1993 Putnam
A6
A6
Part of
1993 Putnam
Problems
(1)
Putnam 1993 A6
Source: 1993 Putnam
10/26/2020
Let
a
0
,
a
1
,
a
2
,
.
.
.
a_0, a_1, a_2, ...
a
0
,
a
1
,
a
2
,
...
be a sequence such that:
a
0
=
2
a_0 = 2
a
0
=
2
; each
a
n
=
2
a_n = 2
a
n
=
2
or
3
;
a
n
=
3; a_n =
3
;
a
n
=
the number of
3
3
3
s between the
n
n
n
th and
n
+
1
n+1
n
+
1
th
2
2
2
in the sequence. So the sequence starts:
233233323332332...
233233323332332 ...
233233323332332...
. Show that we can find
α
\alpha
α
such that
a
n
=
2
a_n = 2
a
n
=
2
iff
n
=
[
α
m
]
n = [\alpha m]
n
=
[
α
m
]
for some integer
m
≥
0
m \geq 0
m
≥
0
.
Putnam