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National and Regional Contests
PEN Problems
PEN K Problems
29
29
Part of
PEN K Problems
Problems
(1)
K 29
Source:
5/25/2007
Find all functions
f
:
Z
∖
{
0
}
→
Q
f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}
f
:
Z
∖
{
0
}
→
Q
such that for all
x
,
y
∈
Z
∖
{
0
}
x,y \in \mathbb{Z}\setminus\{0\}
x
,
y
∈
Z
∖
{
0
}
:
f
(
x
+
y
3
)
=
f
(
x
)
+
f
(
y
)
2
,
x
,
y
∈
Z
∖
{
0
}
f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}
f
(
3
x
+
y
)
=
2
f
(
x
)
+
f
(
y
)
,
x
,
y
∈
Z
∖
{
0
}
function
induction
Functional Equations