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International Contests
Middle European Mathematical Olympiad
2018 Middle European Mathematical Olympiad
1
Determine all functions
Determine all functions
Source: MEMO 2018 I1
September 2, 2018
algebra
function
functional equation
Problem Statement
Let
Q
+
Q^+
Q
+
denote the set of all positive rational number and let
α
∈
Q
+
.
\alpha\in Q^+.
α
∈
Q
+
.
Determine all functions
f
:
Q
+
→
(
α
,
+
∞
)
f:Q^+ \to (\alpha,+\infty )
f
:
Q
+
→
(
α
,
+
∞
)
satisfying
f
(
x
+
y
α
)
=
f
(
x
)
+
f
(
y
)
α
f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}
f
(
α
x
+
y
)
=
α
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
Q
+
.
x,y\in Q^+ .
x
,
y
∈
Q
+
.
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