MathDB
Problems
Contests
International Contests
IMO Longlists
1978 IMO Longlists
29
Finding two functions satisfying conditions
Finding two functions satisfying conditions
Source:
October 28, 2010
function
algebra unsolved
algebra
Problem Statement
Given a nonconstant function
f
:
R
+
⟶
R
f : \mathbb{R}^+ \longrightarrow\mathbb{R}
f
:
R
+
⟶
R
such that
f
(
x
y
)
=
f
(
x
)
f
(
y
)
f(xy) = f(x)f(y)
f
(
x
y
)
=
f
(
x
)
f
(
y
)
for any
x
,
y
>
0
x, y > 0
x
,
y
>
0
, find functions
c
,
s
:
R
+
⟶
R
c, s : \mathbb{R}^+ \longrightarrow \mathbb{R}
c
,
s
:
R
+
⟶
R
that satisfy
c
(
x
y
)
=
c
(
x
)
c
(
y
)
−
s
(
x
)
s
(
y
)
c\left(\frac{x}{y}\right) = c(x)c(y)-s(x)s(y)
c
(
y
x
)
=
c
(
x
)
c
(
y
)
−
s
(
x
)
s
(
y
)
for all
x
,
y
>
0
x, y > 0
x
,
y
>
0
and
c
(
x
)
+
s
(
x
)
=
f
(
x
)
c(x)+s(x) = f(x)
c
(
x
)
+
s
(
x
)
=
f
(
x
)
for all
x
>
0
x > 0
x
>
0
.
Back to Problems
View on AoPS