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IMAR Test
2007 IMAR Test
2
Family of configuration
Family of configuration
Source: IMAR Test 2007
January 28, 2009
geometry
3D geometry
sphere
combinatorics proposed
combinatorics
Problem Statement
Denote by
C
\mathcal{C}
C
the family of all configurations
C
C
C
of
N
>
1
N > 1
N
>
1
distinct points on the sphere
S
2
,
S^2,
S
2
,
and by
H
\mathcal{H}
H
the family of all closed hemispheres
H
H
H
of
S
2
.
S^2.
S
2
.
Compute:
max
H
∈
H
min
C
∈
C
∣
H
∩
C
∣
\displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|
H
∈
H
max
C
∈
C
min
∣
H
∩
C
∣
,
min
H
∈
H
max
C
∈
C
∣
H
∩
C
∣
\displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|
H
∈
H
min
C
∈
C
max
∣
H
∩
C
∣
max
C
∈
C
min
H
∈
H
∣
H
∩
C
∣
\displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|
C
∈
C
max
H
∈
H
min
∣
H
∩
C
∣
and
min
C
∈
C
max
H
∈
H
∣
H
∩
C
∣
.
\displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.
C
∈
C
min
H
∈
H
max
∣
H
∩
C
∣.
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