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Problems
Contests
International Contests
Pan African
2003 Pan African
1
Function N-->N
Function N-->N
Source: Pan African 2003
October 4, 2005
function
logarithms
induction
Problem Statement
Let
N
0
=
{
0
,
1
,
2
⋯
}
N_0=\{0, 1, 2 \cdots \}
N
0
=
{
0
,
1
,
2
⋯
}
. Find all functions:
N
0
→
N
0
N_0 \to N_0
N
0
→
N
0
such that: (1)
f
(
n
)
<
f
(
n
+
1
)
f(n) < f(n+1)
f
(
n
)
<
f
(
n
+
1
)
, all
n
∈
N
0
n \in N_0
n
∈
N
0
; (2)
f
(
2
)
=
2
f(2)=2
f
(
2
)
=
2
; (3)
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn)=f(m)f(n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
, all
m
,
n
∈
N
0
m, n \in N_0
m
,
n
∈
N
0
.
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