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Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2023 Philippine MO
6
Sliding FE
Sliding FE
Source: Philippine MO 2023/6
March 19, 2023
PMO
algebra
functional equation
function
Problem Statement
Find all functions
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that
f
(
2
f
(
x
)
)
=
f
(
x
−
f
(
y
)
)
+
f
(
x
)
+
y
f(2f(x)) = f(x - f(y)) + f(x) + y
f
(
2
f
(
x
))
=
f
(
x
−
f
(
y
))
+
f
(
x
)
+
y
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
.
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