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Chile Classification NMO
2018 Chile Classification NMO Seniors
2018 Chile Classification NMO Seniors
Part of
Chile Classification NMO
Subcontests
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2018 Chile Classification / Qualifying NMO Seniors XXX
p1. From a
1000
1000
1000
-page book, a quantity has been ripped of consecutive of leaves. It is known that the sum of the numbers of the torn pages is
2018
2018
2018
. Determine the numbering of the ripped pages. p2. A square with side
8
8
8
cm is divided into
64
64
64
squares of
1
1
1
cm
2
^2
2
.
7
7
7
little squares are colored black and the rest white. Find the maximum area of a rectangle composed only of small white squares independent of the distribution of the little black squares. p3. Let n
∈
N
\in N
∈
N
. We want to find a partition of
Ω
=
{
1
,
2
,
.
.
.
,
n
}
\Omega = \{1, 2,..., n\}
Ω
=
{
1
,
2
,
...
,
n
}
in
M
M
M
disjoint subsets
S
1
,
S
2
,
.
.
.
,
S
M
S_1, S_2,..., S_M
S
1
,
S
2
,
...
,
S
M
, such that the sum of the elements of each
S
i
S_i
S
i
is the same. What is the maximum value possible of
M
M
M
? p4. In the hypotenuse
B
C
BC
BC
of a right isosceles triangle
A
B
C
ABC
A
BC
are chosen four points
P
1
P_1
P
1
,
P
2
P_2
P
2
,
P
3
P_3
P
3
,
P
4
P_4
P
4
such that
∣
B
P
1
∣
=
∣
P
1
P
2
∣
=
∣
P
2
P
3
∣
=
∣
P
3
P
4
∣
=
∣
P
4
C
∣
|BP_1| = |P_1P_2| = |P_2P_3| = |P_3P_4| = |P_4C|
∣
B
P
1
∣
=
∣
P
1
P
2
∣
=
∣
P
2
P
3
∣
=
∣
P
3
P
4
∣
=
∣
P
4
C
∣
. Choose a point
D
D
D
on leg
A
B
AB
A
B
such that
5
∣
A
D
∣
=
∣
A
B
∣
5|AD| = |AB|
5∣
A
D
∣
=
∣
A
B
∣
. Calculate the sum of the four angles
∠
A
P
i
D
\angle AP_iD
∠
A
P
i
D
,
i
=
1
,
.
.
.
,
4
i = 1,...,4
i
=
1
,
...
,
4
.PS. Seniors p1, p2 were posted as [url=https://artofproblemsolving.com/community/c4h2690917p23356802]Juniors p3,p2 respectively.