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Problems
Contests
National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2016 NZMOC Camp Selection Problems
9
9
Part of
2016 NZMOC Camp Selection Problems
Problems
(1)
occasionally periodic n-tuple
Source: New Zealand NZMOC Camp Selection Problems 2016 p9
9/19/2021
An
n
n
n
-tuple
(
a
1
,
a
2
.
.
.
,
a
n
)
(a_1, a_2 . . . , a_n)
(
a
1
,
a
2
...
,
a
n
)
is occasionally periodic if there exist a non-negative integer
i
i
i
and a positive integer
p
p
p
satisfying
i
+
2
p
≤
n
i + 2p \le n
i
+
2
p
≤
n
and
a
i
+
j
=
a
i
+
j
+
p
a_{i+j} = a_{i+j+p}
a
i
+
j
=
a
i
+
j
+
p
for every
j
=
1
,
2
,
.
.
.
,
p
j = 1, 2, . . . , p
j
=
1
,
2
,
...
,
p
. Let
k
k
k
be a positive integer. Find the least positive integer
n
n
n
for which there exists an
n
n
n
-tuple
(
a
1
,
a
2
.
.
.
,
a
n
)
(a_1, a_2 . . . , a_n)
(
a
1
,
a
2
...
,
a
n
)
with elements from the set
{
1
,
2
,
.
.
.
,
k
}
\{1, 2, . . . , k\}
{
1
,
2
,
...
,
k
}
, which is not occasionally periodic but whose arbitrary extension
(
a
1
,
a
2
,
.
.
.
,
a
n
,
a
n
+
1
)
(a_1, a_2, . . . , a_n, a_{n+1})
(
a
1
,
a
2
,
...
,
a
n
,
a
n
+
1
)
is occasionally periodic for any
a
n
+
1
∈
{
1
,
2
,
.
.
.
,
k
}
a_{n+1} \in \{1, 2, . . . , k\}
a
n
+
1
∈
{
1
,
2
,
...
,
k
}
.
combinatorics
periodic