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National and Regional Contests
PEN Problems
PEN K Problems
19
19
Part of
PEN K Problems
Problems
(1)
K 19
Source:
5/25/2007
Find all functions
f
:
Q
+
→
Q
+
f: \mathbb{Q}^{+}\to \mathbb{Q}^{+}
f
:
Q
+
→
Q
+
such that for all
x
,
y
∈
Q
x,y \in \mathbb{Q}
x
,
y
∈
Q
:
f
(
x
+
y
x
)
=
f
(
x
)
+
f
(
y
)
f
(
x
)
+
2
y
,
x
,
y
∈
Q
+
.
f \left( x+\frac{y}{x}\right) =f(x)+\frac{f(y)}{f(x)}+2y, \; x,y \in \mathbb{Q}^{+}.
f
(
x
+
x
y
)
=
f
(
x
)
+
f
(
x
)
f
(
y
)
+
2
y
,
x
,
y
∈
Q
+
.
function
Functional Equations