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National and Regional Contests
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PEN K Problems
33
K 33
K 33
Source:
May 25, 2007
function
induction
strong induction
algebra
functional equation
Functional Equations
Problem Statement
Find all functions
f
:
Q
→
Q
f: \mathbb{Q}\to \mathbb{Q}
f
:
Q
→
Q
such that for all
x
,
y
,
z
∈
Q
x,y,z \in \mathbb{Q}
x
,
y
,
z
∈
Q
:
f
(
x
+
y
+
z
)
+
f
(
x
−
y
)
+
f
(
y
−
z
)
+
f
(
z
−
x
)
=
3
f
(
x
)
+
3
f
(
y
)
+
3
f
(
z
)
.
f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).
f
(
x
+
y
+
z
)
+
f
(
x
−
y
)
+
f
(
y
−
z
)
+
f
(
z
−
x
)
=
3
f
(
x
)
+
3
f
(
y
)
+
3
f
(
z
)
.
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