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Israel Olympic Revenge
2020 Israel Olympic Revenge
P1
P1
Part of
2020 Israel Olympic Revenge
Problems
(1)
Finite preimages, symmetry
Source: 2020 Israel Olympic Revenge P1
6/11/2022
Find all functions
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
such that for all
x
,
y
∈
R
x,y\in \mathbb{R}
x
,
y
∈
R
one has
f
(
f
(
x
)
+
y
)
=
f
(
x
+
f
(
y
)
)
f(f(x)+y)=f(x+f(y))
f
(
f
(
x
)
+
y
)
=
f
(
x
+
f
(
y
))
and in addition the set
f
−
1
(
a
)
=
{
b
∈
R
∣
f
(
b
)
=
a
}
f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}
f
−
1
(
a
)
=
{
b
∈
R
∣
f
(
b
)
=
a
}
is a finite set for all
a
∈
R
a\in \mathbb{R}
a
∈
R
.
functional equation
algebra
olympic revenge
function
symmetry