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International Contests
IberoAmerican
2009 IberoAmerican
2009 IberoAmerican
Part of
IberoAmerican
Subcontests
(6)
6
1
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Coloring 6000 points using 10 colors
Six thousand points are marked on a circle, and they are colored using 10 colors in such a way that within every group of 100 consecutive points all the colors are used. Determine the least positive integer
k
k
k
with the following property: In every coloring satisfying the condition above, it is possible to find a group of
k
k
k
consecutive points in which all the colors are used.
5
1
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Every rational appears on the sequence exactly once
Consider the sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\geq1}
{
a
n
}
n
≥
1
defined as follows: a_1 \equal{} 1, a_{2k} \equal{} 1 \plus{} a_k and a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}} for every
k
≥
1
k\geq 1
k
≥
1
. Prove that every positive rational number appears on the sequence
{
a
n
}
\{a_n\}
{
a
n
}
exactly once.
4
1
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Circumcircle and external bisector
Given a triangle
A
B
C
ABC
A
BC
of incenter
I
I
I
, let
P
P
P
be the intersection of the external bisector of angle
A
A
A
and the circumcircle of
A
B
C
ABC
A
BC
, and
J
J
J
the second intersection of
P
I
PI
P
I
and the circumcircle of
A
B
C
ABC
A
BC
. Show that the circumcircles of triangles
J
I
B
JIB
J
I
B
and
J
I
C
JIC
J
I
C
are respectively tangent to
I
C
IC
I
C
and
I
B
IB
I
B
.
1
1
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Islands and bridges
Given a positive integer
n
≥
2
n\geq 2
n
≥
2
, consider a set of
n
n
n
islands arranged in a circle. Between every two neigboring islands two bridges are built as shown in the figure. Starting at the island
X
1
X_1
X
1
, in how many ways one can one can cross the
2
n
2n
2
n
bridges so that no bridge is used more than once?
3
1
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Existence of point X and equal circles
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two congruent circles centered at
O
1
O_1
O
1
and
O
2
O_2
O
2
, which intersect at
A
A
A
and
B
B
B
. Take a point
P
P
P
on the arc
A
B
AB
A
B
of
C
2
C_2
C
2
which is contained in
C
1
C_1
C
1
.
A
P
AP
A
P
meets
C
1
C_1
C
1
at
C
C
C
,
C
B
CB
CB
meets
C
2
C_2
C
2
at
D
D
D
and the bisector of
∠
C
A
D
\angle CAD
∠
C
A
D
intersects
C
1
C_1
C
1
and
C
2
C_2
C
2
at
E
E
E
and
L
L
L
, respectively. Let
F
F
F
be the symmetric point of
D
D
D
with respect to the midpoint of
P
E
PE
PE
. Prove that there exists a point
X
X
X
satisfying \angle XFL \equal{} \angle XDC \equal{} 30^\circ and CX \equal{} O_1O_2. Author: Arnoldo Aguilar (El Salvador)
2
1
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Ibero American 2009 Problem 2
Define the succession
a
n
a_{n}
a
n
,
n
>
0
n>0
n
>
0
as n\plus{}m, where
m
m
m
is the largest integer such that
2
2
m
≤
n
2
n
2^{2^{m}}\leq n2^{n}
2
2
m
≤
n
2
n
. Find all numbers that are not in the succession.