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National and Regional Contests
Cuba Contests
Cuba MO
2003 Cuba MO
8
8
Part of
2003 Cuba MO
Problems
(1)
f(uv) = f(u)f(v), f(au) = |a | f(u), f(u) + f(v) <=|u| + |v|, f : C -> R^+
Source: 2003 Cuba MO 2.8
9/15/2024
Find all the functions
f
:
C
→
R
+
f : C \to R^+
f
:
C
→
R
+
such that they fulfill simultaneously the following conditions:
(
i
)
f
(
u
v
)
=
f
(
u
)
f
(
v
)
∀
u
,
v
∈
C
(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C
(
i
)
f
(
uv
)
=
f
(
u
)
f
(
v
)
∀
u
,
v
∈
C
(
i
i
)
f
(
a
u
)
=
∣
a
∣
f
(
u
)
∀
a
∈
R
,
u
∈
C
(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C
(
ii
)
f
(
a
u
)
=
∣
a
∣
f
(
u
)
∀
a
∈
R
,
u
∈
C
(
i
i
i
)
f
(
u
)
+
f
(
v
)
≤
∣
u
∣
+
∣
v
∣
∀
u
,
v
∈
C
(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C
(
iii
)
f
(
u
)
+
f
(
v
)
≤
∣
u
∣
+
∣
v
∣
∀
u
,
v
∈
C
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