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Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2023 Turkey Team Selection Test
7
7
Part of
2023 Turkey Team Selection Test
Problems
(1)
Periodic sequence modulo f(n)
Source: 2023 Turkey TST D3 P7
3/29/2023
Let us call an integer sequence
{
a
1
,
a
2
,
…
}
\{ a_1,a_2, \dots \}
{
a
1
,
a
2
,
…
}
nice if there exist a function
f
:
Z
+
→
Z
+
f: \mathbb{Z^+} \to \mathbb{Z^+}
f
:
Z
+
→
Z
+
such that
a
i
≡
a
j
(
m
o
d
n
)
⟺
i
≡
j
(
m
o
d
f
(
n
)
)
a_i \equiv a_j \pmod{n} \iff i\equiv j \pmod{f(n)}
a
i
≡
a
j
(
mod
n
)
⟺
i
≡
j
(
mod
f
(
n
))
for all
i
,
j
,
n
∈
Z
+
i,j,n \in \mathbb{Z^+}
i
,
j
,
n
∈
Z
+
. Find all nice sequences.
function
number theory