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PAMO Problem 5: Functional equation
Source: 2021 Pan-African Mathematics Olympiad, Problem 5
5/24/2021
Find all functions
f
f
f
:
:
:
R
→
R
\mathbb{R} \rightarrow \mathbb{R}
R
→
R
such that
∀
x
,
y
∈
R
\forall x,y \in \mathbb{R}
∀
x
,
y
∈
R
:
(
f
(
x
)
+
y
)
(
f
(
y
)
+
x
)
=
f
(
x
2
)
+
f
(
y
2
)
+
2
f
(
x
y
)
(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)
(
f
(
x
)
+
y
)
(
f
(
y
)
+
x
)
=
f
(
x
2
)
+
f
(
y
2
)
+
2
f
(
x
y
)
algebra
functional equation
function