MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2004 Chile National Olympiad
2004 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
5
1
Hide problems
on infinite surface of the sea floats a black and bounded oil slick
On the infinite surface of the sea floats a black and bounded oil slick. After every minute the slick and the sea change according to the following law: at each point
P
P
P
of the sea (or of the slick), a disk
D
D
D
of radius
1
1
1
is considered centered on
P
P
P
. If more than half of the area inside the disk
D
D
D
is black, the
P
P
P
point will remain black for the next minute. If more than half of the area inside the disk
D
D
D
is dark blue, the point
P
P
P
will be dark blue for the next minute. In the event that both the clean and the contaminated area within the disk
D
D
D
are the same, its center
P
P
P
will not change color. Can that stain "live" forever or will it disappear at some point?
4
1
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sum of digits of sum of digits of ... sum of digits of 2^{2004}
Take the number
2
2004
2^{2004}
2
2004
and calculate the sum
S
S
S
of all its digits. Then the sum of all the digits of
S
S
S
is calculated to obtain
R
R
R
. Next, the sum of all the digits of
R
R
R
is calculated and so on until a single digit number is reached. Find it. (For example if we take
2
7
=
128
2^7=128
2
7
=
128
, we find that
S
=
11
,
R
=
2
S=11,R=2
S
=
11
,
R
=
2
. So in this case of
2
7
2^7
2
7
the searched digit will be
2
2
2
).
2
1
Hide problems
all points on line, red or blue
Every point on a line is painted either red or blue. Prove that there always exist three points
A
,
B
,
C
A,B,C
A
,
B
,
C
that are painted the same color and are such that the point
B
B
B
is the midpoint of the segment
A
C
AC
A
C
.
1
1
Hide problems
2004 workers with salaries 2000 or 3000$ pesos
A company with
2004
2004
2004
workers celebrated its anniversary by inviting everyone to a lunch served at a round table. When the
2004
2004
2004
workers sat around this table, they formed a circle of people and soon discovered that they all had salaries. different and also that the difference between the salaries of any two neighbors, at the round table, was
2000
2000
2000
or
3000
3000
3000
pesos. Calculate the maximum difference that can exist between the wages of these workers.
6
1
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collinear from Chile, 3 of sides of ABCD are tangent to a circle (2004 L2 p6 )
The
A
B
,
B
C
AB, BC
A
B
,
BC
and
C
D
CD
C
D
segments of the polygon
A
B
C
D
ABCD
A
BC
D
have the same length and are tangent to a circle
S
S
S
, centered on the point
O
O
O
. Let
P
P
P
be the point of tangency of
B
C
BC
BC
with
S
S
S
, and let
Q
Q
Q
be the intersection point of lines
A
C
AC
A
C
and
B
D
BD
B
D
. Show that the point
Q
Q
Q
is collinear with the points
P
P
P
and
O
O
O
.
3
1
Hide problems
given perimeter and length of diagonal in ABCD, Chile 1999 L2 P3
The perimeter, that is, the sum of the lengths of all sides of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, is equal to
2004
2004
2004
meters; while the length of its diagonal
A
C
AC
A
C
is equal to
1001
1001
1001
meters. Find out if the length of the other diagonal
B
D
BD
B
D
can: a) To be equal to only one meter. b) Be equal to the length of the diagonal
A
C
AC
A
C
.