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National and Regional Contests
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PEN O Problems
21
O 21
O 21
Source:
May 25, 2007
Problem Statement
A sequence of integers
a
1
,
a
2
,
a
3
,
⋯
a_{1}, a_{2}, a_{3}, \cdots
a
1
,
a
2
,
a
3
,
⋯
is defined as follows:
a
1
=
1
a_{1}=1
a
1
=
1
, and for
n
≥
1
n \ge 1
n
≥
1
,
a
n
+
1
a_{n+1}
a
n
+
1
is the smallest integer greater than
a
n
a_{n}
a
n
such that
a
i
+
a
j
≠
3
a
k
a_{i}+a_{j} \neq 3a_{k}
a
i
+
a
j
=
3
a
k
for any
i
,
j
,
i, j,
i
,
j
,
and
k
k
k
in
{
1
,
2
,
3
,
⋯
,
n
+
1
}
\{1, 2, 3, \cdots, n+1 \}
{
1
,
2
,
3
,
⋯
,
n
+
1
}
, not necessarily distinct. Determine
a
1998
a_{1998}
a
1998
.
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