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National and Regional Contests
The Philippines Contests
Philippine MO
2019 Philippine MO
1
1
Part of
2019 Philippine MO
Problems
(1)
f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)
Source: 21st Philippine Mathematical Olympiad 2019 p1
1/7/2020
Find all functions
f
:
R
ā
R
f : R \to R
f
:
R
ā
R
such that
f
(
2
x
y
)
+
f
(
f
(
x
+
y
)
)
=
x
f
(
y
)
+
y
f
(
x
)
+
f
(
x
+
y
)
f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)
f
(
2
x
y
)
+
f
(
f
(
x
+
y
))
=
x
f
(
y
)
+
y
f
(
x
)
+
f
(
x
+
y
)
for all real numbers
x
x
x
and
y
y
y
.
functional equation
algebra