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Problems
Contests
International Contests
Cono Sur Olympiad
2010 Cono Sur Olympiad
2010 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
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Infinite sequence of integers
Determine if there exists an infinite sequence
a
0
,
a
1
,
a
2
,
a
3
,
.
.
.
a_0, a_1, a_2, a_3,...
a
0
,
a
1
,
a
2
,
a
3
,
...
of nonegative integers that satisfies the following conditions: (i) All nonegative integers appear in the sequence exactly once. (ii) The succession
b
n
=
a
n
+
n
,
b_n=a_{n}+n,
b
n
=
a
n
+
n
,
,
n
≥
0
n\geq0
n
≥
0
, is formed by all prime numbers and each one appears exactly once.
5
1
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Triangle inscribed in another triangle
The incircle of triangle
A
B
C
ABC
A
BC
touches sides
B
C
BC
BC
,
A
C
AC
A
C
, and
A
B
AB
A
B
at
D
,
E
D, E
D
,
E
, and
F
F
F
respectively. Let
ω
a
,
ω
b
\omega_a, \omega_b
ω
a
,
ω
b
and
ω
c
\omega_c
ω
c
be the circumcircles of triangles
E
A
F
,
D
B
F
EAF, DBF
E
A
F
,
D
BF
, and
D
C
E
DCE
D
CE
, respectively. The lines
D
E
DE
D
E
and
D
F
DF
D
F
cut
ω
a
\omega_a
ω
a
at
E
a
≠
E
E_a\neq{E}
E
a
=
E
and
F
a
≠
F
F_a\neq{F}
F
a
=
F
, respectively. Let
r
A
r_A
r
A
be the line
E
a
F
a
E_{a}F_a
E
a
F
a
. Let
r
B
r_B
r
B
and
r
C
r_C
r
C
be defined analogously. Show that the lines
r
A
r_A
r
A
,
r
B
r_B
r
B
, and
r
C
r_C
r
C
determine a triangle with its vertices on the sides of triangle
A
B
C
ABC
A
BC
.
3
1
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Cutting a polygon
Let us define cutting a convex polygon with
n
n
n
sides by choosing a pair of consecutive sides
A
B
AB
A
B
and
B
C
BC
BC
and substituting them by three segments
A
M
,
M
N
AM, MN
A
M
,
MN
, and
N
C
NC
NC
, where
M
M
M
is the midpoint of
A
B
AB
A
B
and
N
N
N
is the midpoint of
B
C
BC
BC
. In other words, the triangle
M
B
N
MBN
MBN
is removed and a convex polygon with
n
+
1
n+1
n
+
1
sides is obtained. Let
P
6
P_6
P
6
be a regular hexagon with area
1
1
1
.
P
6
P_6
P
6
is cut and the polygon
P
7
P_7
P
7
is obtained. Then
P
7
P_7
P
7
is cut in one of seven ways and polygon
P
8
P_8
P
8
is obtained, and so on. Prove that, regardless of how the cuts are made, the area of
P
n
P_n
P
n
is always greater than
2
/
3
2/3
2/3
.
2
1
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Crickets on a number line
On a line,
44
44
44
points are marked and numbered
1
,
2
,
3
,
…
,
44
1, 2, 3,…,44
1
,
2
,
3
,
…
,
44
from left to right. Various crickets jump around the line. Each starts at point
1
1
1
, jumping on the marked points and ending up at point
44
44
44
. In addition, each cricket jumps from a marked point to another marked point with a greater number. When all the crickets have finished jumping, it turns out that for pair
i
,
j
i, j
i
,
j
with
1
≤
i
<
j
≤
44
{1}\leq{i}<{j}\leq{44}
1
≤
i
<
j
≤
44
, there was a cricket that jumped directly from point
i
i
i
to point
j
j
j
, without visiting any of the points in between the two. Determine the smallest number of crickets such that this is possible.
1
1
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Sum of two fractions
Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:[*]The sum of the fractions is equal to
2
2
2
. [*]The sum of the numerators of the fractions is equal to
1000
1000
1000
.In how many ways can Pedro do this?
4
1
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Cono Sur Olympiad 2010, Problem 4
Pablo and Silvia play on a
2010
×
2010
2010 \times 2010
2010
×
2010
board. To start the game, Pablo writes an integer in every cell. After he is done, Silvia repeats the following operation as many times as she wants: she chooses three cells that form an
L
L
L
, like in the figure below, and adds
1
1
1
to each of the numbers in these three cells. Silvia wins if, after doing the operation many times, all of the numbers in the board are multiples of
10
10
10
. Prove that Silvia can always win.\begin{array}{|c|c} \cline{1-1} \; & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \end{array} \qquad \begin{array}{c|c|} \cline{2-2} \; & \; \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \end{array} \qquad \begin{array}{|c|c} \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \cline{1-1} \end{array} \qquad \begin{array}{c|c|} \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \cline{2-2} \end{array}