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Vojtěch Jarník IMC
2004 VJIMC
Problem 2
FE, R≥0 x R≥0 -> R≥0
FE, R≥0 x R≥0 -> R≥0
Source: VJIMC 2004 2.2
July 12, 2021
fe
functional equation
algebra
Problem Statement
Find all functions
f
:
R
≥
0
×
R
≥
0
→
R
≥
0
f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}
f
:
R
≥
0
×
R
≥
0
→
R
≥
0
such that
1
1
1
.
f
(
x
,
0
)
=
f
(
0
,
x
)
=
x
f(x,0)=f(0,x)=x
f
(
x
,
0
)
=
f
(
0
,
x
)
=
x
for all
x
∈
R
≥
0
x\in\mathbb R_{\ge0}
x
∈
R
≥
0
,
2
2
2
.
f
(
f
(
x
,
y
)
,
z
)
=
f
(
x
,
f
(
y
,
z
)
)
f(f(x,y),z)=f(x,f(y,z))
f
(
f
(
x
,
y
)
,
z
)
=
f
(
x
,
f
(
y
,
z
))
for all
x
,
y
,
z
∈
R
≥
0
x,y,z\in\mathbb R_{\ge0}
x
,
y
,
z
∈
R
≥
0
and
3
3
3
. there exists a real
k
k
k
such that
f
(
x
+
y
,
x
+
z
)
=
k
x
+
f
(
y
,
z
)
f(x+y,x+z)=kx+f(y,z)
f
(
x
+
y
,
x
+
z
)
=
k
x
+
f
(
y
,
z
)
for all
x
,
y
,
z
∈
R
≥
0
x,y,z\in\mathbb R_{\ge0}
x
,
y
,
z
∈
R
≥
0
.
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