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Thailand National Olympiad
2022 Thailand Mathematical Olympiad
5
Two Variables Functions in a Cartesian plane
Two Variables Functions in a Cartesian plane
Source: 2022 Thailand MO Day 1 P5
June 15, 2022
function
functional equation
geometry
analytic geometry
Problem Statement
Determine all functions
f
:
R
×
R
→
R
f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}
f
:
R
×
R
→
R
that satisfies the equation
f
(
x
+
y
+
z
3
,
a
+
b
+
c
3
)
=
f
(
x
,
a
)
f
(
y
,
b
)
f
(
z
,
c
)
f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)
f
(
3
x
+
y
+
z
,
3
a
+
b
+
c
)
=
f
(
x
,
a
)
f
(
y
,
b
)
f
(
z
,
c
)
for any real numbers
x
,
y
,
z
,
a
,
b
,
c
x,y,z,a,b,c
x
,
y
,
z
,
a
,
b
,
c
such that
a
z
+
b
x
+
c
y
≠
a
y
+
b
z
+
c
x
az+bx+cy\neq ay+bz+cx
a
z
+
b
x
+
cy
=
a
y
+
b
z
+
c
x
.
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