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2022 China Team Selection Test
3
Novel FE with multiset condition
Novel FE with multiset condition
Source: 2022 China TST, Test 4 P3
April 30, 2022
functional equation
algebra
Problem Statement
Find all functions
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
such that for any
x
,
y
∈
R
x,y \in \mathbb R
x
,
y
∈
R
, the multiset
{
(
f
(
x
f
(
y
)
+
1
)
,
f
(
y
f
(
x
)
−
1
)
}
\{(f(xf(y)+1),f(yf(x)-1)\}
{(
f
(
x
f
(
y
)
+
1
)
,
f
(
y
f
(
x
)
−
1
)}
is identical to the multiset
{
x
f
(
f
(
y
)
)
+
1
,
y
f
(
f
(
x
)
)
−
1
}
\{xf(f(y))+1,yf(f(x))-1\}
{
x
f
(
f
(
y
))
+
1
,
y
f
(
f
(
x
))
−
1
}
.Note: The multiset
{
a
,
b
}
\{a,b\}
{
a
,
b
}
is identical to the multiset
{
c
,
d
}
\{c,d\}
{
c
,
d
}
if and only if
a
=
c
,
b
=
d
a=c,b=d
a
=
c
,
b
=
d
or
a
=
d
,
b
=
c
a=d,b=c
a
=
d
,
b
=
c
.
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