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Problems
Contests
International Contests
Cono Sur Olympiad
1995 Cono Sur Olympiad
1995 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(3)
2
2
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Segments
There are ten points marked on a circumference, numbered from
1
1
1
to
10
10
10
and join all points with segments. I color the segments, with red someones and others with blue. Without changing the colors of the segments, renumber all the points from the
1
1
1
to the
10
10
10
. Will be possible to color the segments and to renumber the points so that those numbers that were jointed with red are jointed now with blue and the numbers that were jointed with blue they are jointed now with red?
Find the area
The semicircle with centre
O
O
O
and the diameter
A
C
AC
A
C
is divided in two arcs
A
B
AB
A
B
and
B
C
BC
BC
with ratio
1
:
3
1: 3
1
:
3
.
M
M
M
is the midpoint of the radium
O
C
OC
OC
. Let
T
T
T
be the point of arc
B
C
BC
BC
such that the area of the cuadrylateral
O
B
T
M
OBTM
OBTM
is maximum. Find such area in fuction of the radium.
3
2
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Circles inside a rectangle
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with:
A
B
=
a
AB=a
A
B
=
a
,
B
C
=
b
BC=b
BC
=
b
. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides
A
B
AB
A
B
and
A
D
AD
A
D
,the other is tangent to the sides
C
B
CB
CB
and
C
D
CD
C
D
. 1. Find the distance between the centers of the circles(using
a
a
a
and
b
b
b
). 2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.
F(n)
Let
n
n
n
be a natural number and
f
(
n
)
=
2
n
−
1995
⌊
n
1000
⌋
f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor
f
(
n
)
=
2
n
−
1995
⌊
1000
n
⌋
(
⌊
\lfloor
⌊
⌋
\rfloor
⌋
denotes the floor function). 1. Show that if for some integer
r
r
r
:
f
(
f
(
f
.
.
.
f
(
n
)
.
.
.
)
)
=
1995
f(f(f...f(n)...))=1995
f
(
f
(
f
...
f
(
n
)
...
))
=
1995
(where the function
f
f
f
is applied
r
r
r
times), then
n
n
n
is multiple of
1995
1995
1995
. 2. Show that if
n
n
n
is multiple of 1995, then there exists r such that:
f
(
f
(
f
.
.
.
f
(
n
)
.
.
.
)
)
=
1995
f(f(f...f(n)...))=1995
f
(
f
(
f
...
f
(
n
)
...
))
=
1995
(where the function
f
f
f
is applied
r
r
r
times). Determine
r
r
r
if
n
=
1995.500
=
997500
n=1995.500=997500
n
=
1995.500
=
997500
1
2
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Find the number
Find a number with
3
3
3
digits, knowing that the sum of its digits is
9
9
9
, their product is
24
24
24
and also the number read from right to left is
27
38
\frac{27}{38}
38
27
of the original.
Cono sur olympiad 1995
We write the digits of
1995
1995
1995
in the following way:
199511999955111999999555......
199511999955111999999555......
199511999955111999999555......
1. Determine how many digits we have to write such that the sum of the written digits is
2880
2880
2880
. 2.Which digit is in position number
1995
1995
1995
?