MathDB
Problems
Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2019 239 Open Mathematical Olympiad
6
Two conditions for f(x)
Two conditions for f(x)
Source: 239 2019 S6
July 31, 2020
algebra
functional equation
Problem Statement
Find all functions
f
:
(
0
,
+
∞
)
→
R
f : (0, +\infty) \to \mathbb{R}
f
:
(
0
,
+
∞
)
→
R
satisfying the following conditions:
(
i
)
(i)
(
i
)
f
(
x
)
+
f
(
1
x
)
=
1
f(x) + f(\frac{1}{x}) = 1
f
(
x
)
+
f
(
x
1
)
=
1
for all
x
>
0
x> 0
x
>
0
;
(
i
i
)
(ii)
(
ii
)
f
(
x
y
+
x
+
y
)
=
f
(
x
)
f
(
y
)
f(xy + x + y) = f(x)f(y)
f
(
x
y
+
x
+
y
)
=
f
(
x
)
f
(
y
)
for all
x
,
y
>
0
x, y> 0
x
,
y
>
0
.
Back to Problems
View on AoPS