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National and Regional Contests
China Contests
China Team Selection Test
2019 China Team Selection Test
4
Two-variable function
Two-variable function
Source: 2019 China TST Test 3 P4
March 29, 2019
algebra
functional equation
function
Problem Statement
Find all functions
f
:
R
2
→
R
f: \mathbb{R}^2 \rightarrow \mathbb{R}
f
:
R
2
→
R
, such that 1)
f
(
0
,
x
)
f(0,x)
f
(
0
,
x
)
is non-decreasing ; 2) for any
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
,
f
(
x
,
y
)
=
f
(
y
,
x
)
f(x,y)=f(y,x)
f
(
x
,
y
)
=
f
(
y
,
x
)
; 3) for any
x
,
y
,
z
∈
R
x,y,z \in \mathbb{R}
x
,
y
,
z
∈
R
,
(
f
(
x
,
y
)
−
f
(
y
,
z
)
)
(
f
(
y
,
z
)
−
f
(
z
,
x
)
)
(
f
(
z
,
x
)
−
f
(
x
,
y
)
)
=
0
(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0
(
f
(
x
,
y
)
−
f
(
y
,
z
))
(
f
(
y
,
z
)
−
f
(
z
,
x
))
(
f
(
z
,
x
)
−
f
(
x
,
y
))
=
0
; 4) for any
x
,
y
,
a
∈
R
x,y,a \in \mathbb{R}
x
,
y
,
a
∈
R
,
f
(
x
+
a
,
y
+
a
)
=
f
(
x
,
y
)
+
a
f(x+a,y+a)=f(x,y)+a
f
(
x
+
a
,
y
+
a
)
=
f
(
x
,
y
)
+
a
.
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