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National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2010 Paraguay Mathematical Olympiad
2010 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
5
1
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Paraguayan National Olympiad 2010, Level 3, Problem 5
In a triangle
A
B
C
ABC
A
BC
, let
D
D
D
,
E
E
E
and
F
F
F
be the feet of the altitudes from
A
A
A
,
B
B
B
and
C
C
C
respectively. Let
D
′
D'
D
′
,
E
′
E'
E
′
and
F
′
F'
F
′
be the second intersection of lines
A
D
AD
A
D
,
B
E
BE
BE
and
C
F
CF
CF
with the circumcircle of
A
B
C
ABC
A
BC
. Show that the triangles
D
E
F
DEF
D
EF
and
D
′
E
′
F
′
D'E'F'
D
′
E
′
F
′
are similar.
4
1
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Paraguayan National Olympiad 2010, Level 3, Problem 4
Find all 4-digit numbers
a
b
c
d
‾
\overline{abcd}
ab
c
d
that are multiples of
11
11
11
, such that the 2-digit number
a
c
‾
\overline{ac}
a
c
is a multiple of
7
7
7
and
a
+
b
+
c
+
d
=
d
2
a + b + c + d = d^2
a
+
b
+
c
+
d
=
d
2
.
3
1
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Paraguayan National Olympiad 2010, Level 3, Problem 3
In a triangle
A
B
C
ABC
A
BC
, let
M
M
M
be the midpoint of
A
C
AC
A
C
. If
B
C
=
2
3
M
C
BC = \frac{2}{3} MC
BC
=
3
2
MC
and
∠
B
M
C
=
2
∠
A
B
M
\angle{BMC}=2 \angle{ABM}
∠
BMC
=
2∠
A
BM
, determine
A
M
A
B
\frac{AM}{AB}
A
B
A
M
.
2
1
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Paraguayan National Olympiad 2010, Level 3, Problem 2
A series of figures is shown in the picture below, each one of them created by following a secret rule. If the leftmost figure is considered the first figure, how many squares will the 21st figure have?http://www.artofproblemsolving.com/Forum/download/file.php?id=49934Note: only the little squares are to be counted (i.e., the
2
×
2
2 \times 2
2
×
2
squares,
3
×
3
3 \times 3
3
×
3
squares,
…
\dots
…
should not be counted) Extra (not part of the original problem): How many squares will the 21st figure have, if we consider all
1
×
1
1 \times 1
1
×
1
squares, all
2
×
2
2 \times 2
2
×
2
squares, all
3
×
3
3 \times 3
3
×
3
squares, and so on?.
1
1
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Paraguayan National Olympiad 2010, Level 3, Problem 1
The picture below shows the way Juan wants to divide a square field in three regions, so that all three of them share a well at vertex
B
B
B
. If the side length of the field is
60
60
60
meters, and each one of the three regions has the same area, how far must the points
M
M
M
and
N
N
N
be from
D
D
D
?Note: the area of each region includes the area the well occupies.[asy] pair A=(0,0),B=(60,0),C=(60,-60),D=(0,-60),M=(0,-40),N=(20,-60); pathpen=black; D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle); D(B--MP("M",M,W)); D(B--MP("N",N,S)); D(CR(B,3));[/asy]