MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2022 Iran Team Selection Test
12
Iran TST P12
Iran TST P12
Source: Iranian TST 2022 problem 12
April 2, 2022
algebra
interval
functional equation
Problem Statement
suppose that
A
A
A
is the set of all Closed intervals
[
a
,
b
]
⊂
R
[a,b] \subset \mathbb{R}
[
a
,
b
]
⊂
R
. Find all functions
f
:
R
→
A
f:\mathbb{R} \rightarrow A
f
:
R
→
A
such that
∙
\bullet
∙
x
∈
f
(
y
)
⇔
y
∈
f
(
x
)
x \in f(y) \Leftrightarrow y \in f(x)
x
∈
f
(
y
)
⇔
y
∈
f
(
x
)
∙
\bullet
∙
∣
x
−
y
∣
>
2
⇔
f
(
x
)
∩
f
(
y
)
=
∅
|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing
∣
x
−
y
∣
>
2
⇔
f
(
x
)
∩
f
(
y
)
=
∅
∙
\bullet
∙
For all real numbers
0
≤
r
≤
1
0\leq r\leq 1
0
≤
r
≤
1
,
f
(
r
)
=
[
r
2
−
1
,
r
2
+
1
]
f(r)=[r^2-1,r^2+1]
f
(
r
)
=
[
r
2
−
1
,
r
2
+
1
]
Proposed by Matin Yousefi
Back to Problems
View on AoPS