MathDB
Problems
Contests
International Contests
APMO
2010 APMO
5
5
Part of
2010 APMO
Problems
(1)
Asian Pacific Mathematical Olympiad 2010 Problem 5
Source:
5/7/2010
Find all functions
f
f
f
from the set
R
\mathbb{R}
R
of real numbers into
R
\mathbb{R}
R
which satisfy for all
x
,
y
,
z
∈
R
x, y, z \in \mathbb{R}
x
,
y
,
z
∈
R
the identity
f
(
f
(
x
)
+
f
(
y
)
+
f
(
z
)
)
=
f
(
f
(
x
)
−
f
(
y
)
)
+
f
(
2
x
y
+
f
(
z
)
)
+
2
f
(
x
z
−
y
z
)
.
f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).
f
(
f
(
x
)
+
f
(
y
)
+
f
(
z
))
=
f
(
f
(
x
)
−
f
(
y
))
+
f
(
2
x
y
+
f
(
z
))
+
2
f
(
x
z
−
yz
)
.
function
algebra proposed
algebra