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National and Regional Contests
Romania Contests
IMAR Test
2007 IMAR Test
2007 IMAR Test
Part of
IMAR Test
Subcontests
(3)
2
1
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Family of configuration
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
∣.
3
1
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N integers
Prove that N\geq 2n \minus{} 2 integers, of absolute value not higher than
n
>
2
n > 2
n
>
2
, and of absolute value of their sum
S
S
S
less than n \minus{} 1, there exist some of sum
0.
0.
0.
Show that for |S| \equal{} n \minus{} 1 this is not anymore true, and neither for N \equal{} 2n \minus{} 3 (when even for |S| \equal{} 1 this is not anymore true).
1
1
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sup(S)
For real numbers
x
i
>
1
,
1
≤
i
≤
n
,
n
≥
2
,
x_{i}>1,1\leq i\leq n,n\geq 2,
x
i
>
1
,
1
≤
i
≤
n
,
n
≥
2
,
such that: \frac{x_{i}^2}{x_{i}\minus{}1}\geq S\equal{}\displaystyle\sum^n_{j\equal{}1}x_{j}, for all i\equal{}1,2\dots, n find, with proof,
sup
S
.
\sup S.
sup
S
.