MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2020 China National Olympiad
4
4
Part of
2020 China National Olympiad
Problems
(1)
Common intersection of arcs
Source: 2020 CMO P4
11/27/2019
Find the largest positive constant
C
C
C
such that the following is satisfied: Given
n
n
n
arcs (containing their endpoints)
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots ,A_n
A
1
,
A
2
,
…
,
A
n
on the circumference of a circle, where among all sets of three arcs
(
A
i
,
A
j
,
A
k
)
(A_i,A_j,A_k)
(
A
i
,
A
j
,
A
k
)
(
1
≤
i
<
j
<
k
≤
n
)
(1\le i< j< k\le n)
(
1
≤
i
<
j
<
k
≤
n
)
, at least half of them has
A
i
∩
A
j
∩
A
k
A_i\cap A_j\cap A_k
A
i
∩
A
j
∩
A
k
nonempty, then there exists
l
>
C
n
l>Cn
l
>
C
n
, such that we can choose
l
l
l
arcs among
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots ,A_n
A
1
,
A
2
,
…
,
A
n
, whose intersection is nonempty.
combinatorics