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JOM 2015 Shortlist
N8
N8
Part of
JOM 2015 Shortlist
Problems
(1)
Number of functions
Source: Junior Olympiad of Malaysia Shortlist 2015 N8
7/17/2015
Set
p
≥
5
p\ge 5
p
≥
5
be a prime number and
n
n
n
be a natural number. Let
f
f
f
be a function
f
:
Z
≠
0
→
N
0
f: \mathbb{Z_{ \neq }}_0 \rightarrow \mathbb{ N }_0
f
:
Z
=
0
→
N
0
satisfy the following conditions: i) For all sequences of integers satisfy
a
i
∉
{
0
,
1
}
a_i \not\in \{0, 1\}
a
i
∈
{
0
,
1
}
, and
p
p
p
∤
\not |
∣
a
i
−
1
a_i-1
a
i
−
1
,
∀
\forall
∀
1
≤
i
≤
p
−
2
1 \le i \le p-2
1
≤
i
≤
p
−
2
,\\
∑
i
=
1
p
−
2
f
(
a
i
)
=
f
(
a
1
a
2
⋯
a
p
−
2
)
\displaystyle \sum^{p-2}_{i=1}f(a_i)=f(a_1a_2 \cdots a_{p-2})
i
=
1
∑
p
−
2
f
(
a
i
)
=
f
(
a
1
a
2
⋯
a
p
−
2
)
ii) For all coprime integers
a
a
a
and
b
b
b
,
a
≡
b
(
m
o
d
p
)
⇒
f
(
a
)
=
f
(
b
)
a \equiv b \pmod p \Rightarrow f(a)=f(b)
a
≡
b
(
mod
p
)
⇒
f
(
a
)
=
f
(
b
)
iii) There exist
k
∈
Z
≠
0
k \in \mathbb{Z}_{\neq 0}
k
∈
Z
=
0
that satisfy
f
(
k
)
=
n
f(k)=n
f
(
k
)
=
n
Prove that the number of such functions is
d
(
n
)
d(n)
d
(
n
)
, where
d
(
n
)
d(n)
d
(
n
)
denotes the number of divisors of
n
n
n
.
function
number theory