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National and Regional Contests
Chile Contests
Chile Classification NMO
2008 Chile Classification NMO Seniors
2008 Chile Classification NMO Seniors
Part of
Chile Classification NMO
Subcontests
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2008 Chile Classification / Qualifying NMO Seniors XX
p1. You have
680
680
680
oranges stacked in a triangular pyramid. How many oranges are in the base of the pyramid? p2. Four points are marked on each side of a square with side
5
5
5
cm in order to subdivide each side in five equal parts, and join as in the figure. What is the area of the shaded region? https://cdn.artofproblemsolving.com/attachments/d/b/7756e2af51f6ef3443528e204c2768a2cc4738.jpg p3. The set
Z
2
Z^2
Z
2
of the integers is divided into
n
n
n
(disjoint) parts and not empty
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2, ..., A_n
A
1
,
A
2
,
...
,
A
n
they will verify the following property: if
a
a
a
and
b
b
b
belong to
A
i
A_i
A
i
then their sum
a
+
b
a + b
a
+
b
belongs to the same set
A
i
A_i
A
i
. Determine the possible values of the positive integer
n
n
n
. p4. The sequences
x
n
x_n
x
n
,
y
n
y_n
y
n
are defined by the following rules:
x
0
=
2
,
x
1
=
5
,
x
n
+
1
=
x
n
+
2
x
n
−
1
x_0 = 2, x_1 = 5, x_{n + 1} = xn + 2x_{n-1}
x
0
=
2
,
x
1
=
5
,
x
n
+
1
=
x
n
+
2
x
n
−
1
,
y
0
=
3
,
y
1
=
4
,
y
n
+
1
=
y
n
+
2
y
n
−
1
y_0 = 3, y_1 = 4, y_{n + 1} = y_n + 2y_{n-1}
y
0
=
3
,
y
1
=
4
,
y
n
+
1
=
y
n
+
2
y
n
−
1
. Prove that the sets
{
x
n
:
n
≥
0
}
\{x_n: n\ge 0 \}
{
x
n
:
n
≥
0
}
and
{
y
n
:
n
≥
0
}
\{y_n: n\ge 0\}
{
y
n
:
n
≥
0
}
are disjoint. p5. We have two circles
C
1
C_1
C
1
and
C
2
C_2
C
2
tangent (externally) to each other and tangent to a line
L
L
L
(on the same side). From the point P of greatest height (wrt
L
L
L
) in
C
1
C_1
C
1
the tangent is drawn "top"
P
Q
PQ
PQ
to
C
2
C_2
C
2
: see figure. Prove that the length of
P
Q
PQ
PQ
equals the diameter of
C
1
C_1
C
1
. https://cdn.artofproblemsolving.com/attachments/3/f/0aaf51443ab90b49821c22cbf06936167f2335.jpg p6. In each square of a board
n
×
n
n \times n
n
×
n
there is a light bulb. In addition, there are
2
n
2n
2
n
switches. In each row there is a switch that when pressed, changes the state of the bulbs of that row (those that were on are turned off, and those that are off turn on). In each column also there is a switch that changes the state of the bulbs in it. Using these switches, is it always possible to arrive, from any initial state, to a state in which the number of lit bulbs in each row or column is less than or equal to the number of off bulbs in that row or column?PS. Seniors P1, P2, P3, P6 were posted also as [url=https://artofproblemsolving.com/community/c4h2689997p23346441]Juniors P1,P2, variation of P3, easier version of P6.