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Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2008 South africa National Olympiad
6
6
Part of
2008 South africa National Olympiad
Problems
(1)
Find all pairs of functions
Source: South African MO 2008 Q6
5/27/2012
Find all function pairs
(
f
,
g
)
(f,g)
(
f
,
g
)
where each
f
f
f
and
g
g
g
is a function defined on the integers and with values, such that, for all integers
a
a
a
and
b
b
b
,
f
(
a
+
b
)
=
f
(
a
)
g
(
b
)
+
g
(
a
)
f
(
b
)
g
(
a
+
b
)
=
g
(
a
)
g
(
b
)
ā
f
(
a
)
f
(
b
)
.
f(a+b)=f(a)g(b)+g(a)f(b)\\ g(a+b)=g(a)g(b)-f(a)f(b).
f
(
a
+
b
)
=
f
(
a
)
g
(
b
)
+
g
(
a
)
f
(
b
)
g
(
a
+
b
)
=
g
(
a
)
g
(
b
)
ā
f
(
a
)
f
(
b
)
.
function
trigonometry
quadratics
algebra unsolved
algebra