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Contests
International Contests
Cono Sur Olympiad
2011 Cono Sur Olympiad
2011 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
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Cono Sur Olympiad 2011, Problem 6
Let
Q
Q
Q
be a
(
2
n
+
1
)
×
(
2
n
+
1
)
(2n+1) \times (2n+1)
(
2
n
+
1
)
×
(
2
n
+
1
)
board. Some of its cells are colored black in such a way that every
2
×
2
2 \times 2
2
×
2
board of
Q
Q
Q
has at most
2
2
2
black cells. Find the maximum amount of black cells that the board may have.
5
1
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Cono Sur Olympiad 2011, Problem 5
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
a point in
A
C
AC
A
C
. If
∠
C
B
D
−
∠
A
B
D
=
6
0
∘
,
B
D
C
^
=
3
0
∘
\angle{CBD} - \angle{ABD} = 60^{\circ}, \hat{BDC} = 30^{\circ}
∠
CB
D
−
∠
A
B
D
=
6
0
∘
,
B
D
C
^
=
3
0
∘
and also
A
B
⋅
B
C
=
B
D
2
AB \cdot BC = BD^{2}
A
B
⋅
BC
=
B
D
2
, determine the measure of all the angles of triangle
A
B
C
ABC
A
BC
.
4
1
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Cono Sur Olympiad 2011, Problem 4
A number
a
b
c
d
‾
\overline{abcd}
ab
c
d
is called balanced if
a
+
b
=
c
+
d
a+b=c+d
a
+
b
=
c
+
d
. Find all balanced numbers with 4 digits that are the sum of two palindrome numbers.
3
1
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Cono Sur Olympiad 2011, Problem 3
Let
A
B
C
ABC
A
BC
be an equilateral triangle. Let
P
P
P
be a point inside of it such that the square root of the distance of
P
P
P
to one of the sides is equal to the sum of the square roots of the distances of
P
P
P
to the other two sides. Find the geometric place of
P
P
P
.
2
1
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Cono Sur Olympiad 2011, Problem 2
The numbers
1
1
1
through
4
n
4^{n}
4
n
are written on a board. In each step, Pedro erases two numbers
a
a
a
and
b
b
b
from the board, and writes instead the number
a
b
2
a
2
+
2
b
2
\frac{ab}{\sqrt{2a^2+2b^2}}
2
a
2
+
2
b
2
ab
. Pedro repeats this procedure until only one number remains. Prove that this number is less than
1
n
\frac{1}{n}
n
1
, no matter what numbers Pedro chose in each step.
1
1
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Cono Sur Olympiad 2011, Problem 1
Find all triplets of positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
x
2
+
y
2
+
z
2
=
2011
x^{2}+y^{2}+z^{2}=2011
x
2
+
y
2
+
z
2
=
2011
.