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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2000 Iran MO (3rd Round)
2
Not hard
Not hard
Source: Iran 2000
May 31, 2004
function
number theory proposed
number theory
Problem Statement
Find all f:N
⟶
\longrightarrow
⟶
N that: a)
f
(
m
)
=
1
⟺
m
=
1
f(m)=1 \Longleftrightarrow m=1
f
(
m
)
=
1
⟺
m
=
1
b)
d
=
g
c
d
(
m
,
n
)
f
(
m
⋅
n
)
=
f
(
m
)
⋅
f
(
n
)
f
(
d
)
d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)}
d
=
g
c
d
(
m
,
n
)
f
(
m
⋅
n
)
=
f
(
d
)
f
(
m
)
⋅
f
(
n
)
c)
f
2000
(
m
)
=
f
(
m
)
f^{2000}(m)=f(m)
f
2000
(
m
)
=
f
(
m
)
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