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National and Regional Contests
Indonesia Contests
Indonesia MO Shortlist
2015 Indonesia MO Shortlist
A4
A4
Part of
2015 Indonesia MO Shortlist
Problems
(1)
Sorry, Just too many functions
Source: INAMO 2015 Shortlist A4
12/30/2018
Determine all functions
f
:
R
×
R
→
R
f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}
f
:
R
×
R
→
R
such that
f
(
x
,
y
)
+
f
(
y
,
z
)
+
f
(
z
,
x
)
=
max
{
x
,
y
,
z
}
−
min
{
x
,
y
,
z
}
f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \}
f
(
x
,
y
)
+
f
(
y
,
z
)
+
f
(
z
,
x
)
=
max
{
x
,
y
,
z
}
−
min
{
x
,
y
,
z
}
for every
x
,
y
,
z
∈
R
x,y,z \in \mathbb{R}
x
,
y
,
z
∈
R
and there exists some real
a
a
a
such that
f
(
x
,
a
)
=
f
(
a
,
x
)
f(x,a) = f(a,x)
f
(
x
,
a
)
=
f
(
a
,
x
)
for every
x
∈
R
x \in \mathbb{R}
x
∈
R
.
functions
algebra
functional equation
function