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Vietnam National Olympiad
2007 Vietnam National Olympiad
2
Vietnamese Olympiad national 2007, problem 5
Vietnamese Olympiad national 2007, problem 5
Source:
February 8, 2007
function
limit
algebra unsolved
algebra
Problem Statement
Given a number
b
>
0
b>0
b
>
0
, find all functions
f
:
R
→
R
f: \mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
such that:
f
(
x
+
y
)
=
f
(
x
)
.
3
b
y
+
f
(
y
)
−
1
+
b
x
.
(
3
b
y
+
f
(
y
)
−
1
−
b
y
)
∀
x
,
y
∈
R
f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}
f
(
x
+
y
)
=
f
(
x
)
.
3
b
y
+
f
(
y
)
−
1
+
b
x
.
(
3
b
y
+
f
(
y
)
−
1
−
b
y
)
∀
x
,
y
∈
R
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