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National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2023 Switzerland - Final Round
5
5
Part of
2023 Switzerland - Final Round
Problems
(1)
Functional equation with -1 missing from domain
Source: Swiss MO 2023/5
3/12/2023
Let
D
D
D
be the set of real numbers excluding
−
1
-1
−
1
. Find all functions
f
:
D
→
D
f: D \to D
f
:
D
→
D
such that for all
x
,
y
∈
D
x,y \in D
x
,
y
∈
D
satisfying
x
≠
0
x \neq 0
x
=
0
and
y
≠
−
x
y \neq -x
y
=
−
x
, the equality
(
f
(
f
(
x
)
)
+
y
)
f
(
y
x
)
+
f
(
f
(
y
)
)
=
x
(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x
(
f
(
f
(
x
))
+
y
)
f
(
x
y
)
+
f
(
f
(
y
))
=
x
holds.
algebra
functional equation
function
domain