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Taiwan Mathematics Olympiad
2024 Taiwan Mathematics Olympiad
3
3
Part of
2024 Taiwan Mathematics Olympiad
Problems
(1)
$2f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2$
Source: 2024 Taiwan Mathematics Olympiad
2/4/2024
Find all functions
f
f
f
from real numbers to real numbers such that
2
f
(
(
x
+
y
)
2
)
=
f
(
x
+
y
)
+
(
f
(
x
)
)
2
+
(
4
y
−
1
)
f
(
x
)
−
2
y
+
4
y
2
2f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2
2
f
((
x
+
y
)
2
)
=
f
(
x
+
y
)
+
(
f
(
x
)
)
2
+
(
4
y
−
1
)
f
(
x
)
−
2
y
+
4
y
2
holds for all real numbers
x
x
x
and
y
y
y
.
algebra
functional equation
Taiwan