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Problems
Contests
International Contests
Cono Sur Olympiad
2012 Cono Sur Olympiad
2012 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
5
1
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Cono Sur Olympiad 2012
5.
A
A
A
and
B
B
B
play alternating turns on a
2012
×
2013
2012 \times 2013
2012
×
2013
board with enough pieces of the following types:Type
1
1
1
: Piece like Type
2
2
2
but with one square at the right of the bottom square. Type
2
2
2
: Piece of
2
2
2
consecutive squares, one over another. Type
3
3
3
: Piece of
1
1
1
square.At his turn,
A
A
A
must put a piece of the type
1
1
1
on available squares of the board.
B
B
B
, at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If
A
A
A
starts playing, decide who has a winning strategy.Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types
1
1
1
,
2
2
2
and
3
3
3
can be put on exactly
3
3
3
,
2
2
2
and
1
1
1
squares of the board respectively.
6
1
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Cono Sur Olympiad 2012
6. Consider a triangle
A
B
C
ABC
A
BC
with
1
<
A
B
A
C
<
3
2
1 < \frac{AB}{AC} < \frac{3}{2}
1
<
A
C
A
B
<
2
3
. Let
M
M
M
and
N
N
N
, respectively, be variable points of the sides
A
B
AB
A
B
and
A
C
AC
A
C
, different from
A
A
A
, such that
M
B
A
C
−
N
C
A
B
=
1
\frac{MB}{AC} - \frac{NC}{AB} = 1
A
C
MB
−
A
B
NC
=
1
. Show that circumcircle of triangle
A
M
N
AMN
A
MN
pass through a fixed point different from
A
A
A
.
1
1
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Cono Sur Olympiad 2012
1. Around a circumference are written
2012
2012
2012
number, each of with is equal to
1
1
1
or
−
1
-1
−
1
. If there are not
10
10
10
consecutive numbers that sum
0
0
0
, find all possible values of the sum of the
2012
2012
2012
numbers.
2
1
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Cono Sur Olympiad 2012
2. In a square
A
B
C
D
ABCD
A
BC
D
, let
P
P
P
be a point in the side
C
D
CD
C
D
, different from
C
C
C
and
D
D
D
. In the triangle
A
B
P
ABP
A
BP
, the altitudes
A
Q
AQ
A
Q
and
B
R
BR
BR
are drawn, and let
S
S
S
be the intersection point of lines
C
Q
CQ
CQ
and
D
R
DR
D
R
. Show that
∠
A
S
B
=
90
\angle ASB=90
∠
A
SB
=
90
.
3
1
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Cono Sur Olympiad 2012
3. Show that there do not exist positive integers
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
, pairwise co-prime, such that
a
b
+
c
d
ab+cd
ab
+
c
d
,
a
c
+
b
d
ac+bd
a
c
+
b
d
and
a
d
+
b
c
ad+bc
a
d
+
b
c
are odd divisors of the number
(
a
+
b
−
c
−
d
)
(
a
−
b
+
c
−
d
)
(
a
−
b
−
c
+
d
)
(a+b-c-d)(a-b+c-d)(a-b-c+d)
(
a
+
b
−
c
−
d
)
(
a
−
b
+
c
−
d
)
(
a
−
b
−
c
+
d
)
.
4
1
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Cono Sur Olympiad 2012
4. Find the biggest positive integer
n
n
n
, lesser thar
2012
2012
2012
, that has the following property: If
p
p
p
is a prime divisor of
n
n
n
, then
p
2
−
1
p^2 - 1
p
2
−
1
is a divisor of
n
n
n
.