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Mathematical Excellence Olympiad
2019 IMEO
3
Challenging Functional Equation
Challenging Functional Equation
Source: IMEO 2019, Problem 3
October 14, 2019
Problem Statement
Find all functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that for all real
x
,
y
x, y
x
,
y
, the following relation holds:
(
x
+
y
)
⋅
f
(
x
+
y
)
=
f
(
f
(
x
)
+
y
)
⋅
f
(
x
+
f
(
y
)
)
.
(x+y) \cdot f(x+y)= f(f(x)+y) \cdot f(x+f(y)).
(
x
+
y
)
⋅
f
(
x
+
y
)
=
f
(
f
(
x
)
+
y
)
⋅
f
(
x
+
f
(
y
))
.
Proposed by Vadym Koval (Ukraine)
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