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National and Regional Contests
Saudi Arabia Contests
Saudi Arabia GMO TST
2013 Saudi Arabia GMO TST
1
f(x)f(y) = f(xy) + f (x/y)
f(x)f(y) = f(xy) + f (x/y)
Source: 2013 Saudi Arabia GMO TST III p1
July 26, 2020
functional
algebra
functional equation
Problem Statement
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
which satisfy
f
(
3
3
x
)
=
3
f
(
x
)
−
2
3
3
x
f \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x
f
(
3
3
x
)
=
3
f
(
x
)
−
3
2
3
x
and
f
(
x
)
f
(
y
)
=
f
(
x
y
)
+
f
(
x
y
)
f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right)
f
(
x
)
f
(
y
)
=
f
(
x
y
)
+
f
(
y
x
)
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
, with
y
≠
0
y \ne 0
y
=
0
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