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Problems
Contests
National and Regional Contests
Chile Contests
Chile Classification NMO
2000 Chile Classification NMO Seniors
2000 Chile Classification NMO Seniors
Part of
Chile Classification NMO
Subcontests
(1)
1
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2000 Chile Classification / Qualifying NMO Seniors XII
p1. Prove that it is possible to color, with
5
5
5
colors, the cells of a checkered board of
2000
×
2000
2000 \times 2000
2000
×
2000
, so that the colors of any square and its four neighbors are all different (neighbor means a box that has one side in common with another). p2. There are
25
25
25
pieces of cheese, all of different weight. Decide if it is always possible to cut one of the pieces in two parts and then place the
26
26
26
pieces in
2
2
2
packages such that the two packages they weigh the most and each of the parts that are cut , are in different packages. p3. A square
A
B
C
D
ABCD
A
BC
D
, with center
O
O
O
and side
1
1
1
, rotates an angle
α
\alpha
α
about
O
O
O
. Determine the common area of the original square and the resulting square. p4. Prove that
∑
n
=
1
2000
1
n
3
+
3
n
2
+
2
n
<
1
4
\sum_{n = 1}^{2000}\frac{1}{n^3 + 3n^2 + 2n}< \frac14
n
=
1
∑
2000
n
3
+
3
n
2
+
2
n
1
<
4
1
p5. Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence with the following properties:
∙
\bullet
∙
a
0
=
1
,
a
1
=
4
a_0 = 1, a_1 = 4
a
0
=
1
,
a
1
=
4
∙
\bullet
∙
a
n
+
2
=
2
a
n
+
1
−
a
n
a_{n+2} = 2a_{n + 1}-a_n
a
n
+
2
=
2
a
n
+
1
−
a
n
∙
\bullet
∙
a
2
n
+
2
=
2
a
n
−
a
n
+
1
a_{2n + 2} = 2a_n-a_{n + 1}
a
2
n
+
2
=
2
a
n
−
a
n
+
1
Prove that the number
2000
2000
2000
is not in the sequence. p6. Let
P
P
P
be a part of the plane with an area greater than
1
1
1
.
∙
\bullet
∙
Prove that there are two points
(
x
1
,
y
1
)
(x_1, y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2, y_2)
(
x
2
,
y
2
)
in
P
P
P
such that the values of
(
x
1
,
y
1
)
(x_1, y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2, y_2)
(
x
2
,
y
2
)
are both integers.
∙
\bullet
∙
Is the above statement true if the area of
P
P
P
is
1
1
1
?Note:
P
P
P
is not necessarily connected, that is, it can be made up of several more small ''parts''. In such a case, we say that its area is the sum of the areas of its ''parts''. p7. In Chile Chico, the Andean, Central and Baha peoples participate in the
X
X
X
-tlon Olympic, consisting of
X
X
X
tests to be disputed. In each test, the score for first place is higher than the score for second place, and the latter is higher than for third place. Also, the score, and the scores are positive integers. The summed
X
X
X
-tlon scores were:
∙
\bullet
∙
Andean People:
22
22
22
pts.
∙
\bullet
∙
Pueblo de la Baha:
15
15
15
pts.
∙
\bullet
∙
Central Town:
14
14
14
pts. If it is known that the Central people won the shooting: a) How many tests were disputed? b) Who came out second in the long jump?