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National and Regional Contests
Vietnam Contests
Pre - Vietnam Mathematical Olympiad
2011 Pre - Vietnam Mathematical Olympiad
2
System of functional equations
System of functional equations
Source: Pre-VMO 2012 - Problem 6
November 27, 2011
function
algebra proposed
algebra
Problem Statement
Find all function
f
,
g
:
Q
→
Q
f,g: \mathbb{Q} \to \mathbb{Q}
f
,
g
:
Q
→
Q
such that
f
(
g
(
x
)
−
g
(
y
)
)
=
f
(
g
(
x
)
)
−
y
g
(
f
(
x
)
−
f
(
y
)
)
=
g
(
f
(
x
)
)
−
y
\begin{array}{l} f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y \\ g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y \\ \end{array}
f
(
g
(
x
)
−
g
(
y
)
)
=
f
(
g
(
x
)
)
−
y
g
(
f
(
x
)
−
f
(
y
)
)
=
g
(
f
(
x
)
)
−
y
for all
x
,
y
∈
Q
x,y \in \mathbb{Q}
x
,
y
∈
Q
.
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