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National and Regional Contests
Malaysia Contests
JOM Shortlists
JOM 2015 Shortlist
N7
N7
Part of
JOM 2015 Shortlist
Problems
(1)
Function
Source: Junior Olympiad of Malaysia Shortlist 2015 N7
7/17/2015
Find all functions
f
:
N
→
N
0
f:\mathbb{N} \rightarrow \mathbb{ N }_0
f
:
N
→
N
0
satisfy the following conditions:i)
f
(
a
b
)
=
f
(
a
)
+
f
(
b
)
−
f
(
gcd
(
a
,
b
)
)
,
∀
a
,
b
∈
N
f(ab)=f(a)+f(b)-f(\gcd(a,b)), \forall a,b \in \mathbb{N}
f
(
ab
)
=
f
(
a
)
+
f
(
b
)
−
f
(
g
cd
(
a
,
b
))
,
∀
a
,
b
∈
N
ii) For all primes
p
p
p
and natural numbers
a
a
a
,
f
(
a
)
≥
f
(
a
p
)
⇒
f
(
a
)
+
f
(
p
)
≥
f
(
a
)
f
(
p
)
+
1
f(a)\ge f(ap) \Rightarrow f(a)+f(p) \ge f(a)f(p)+1
f
(
a
)
≥
f
(
a
p
)
⇒
f
(
a
)
+
f
(
p
)
≥
f
(
a
)
f
(
p
)
+
1
function
number theory