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National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2003 Paraguay Mathematical Olympiad
2003 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
4
1
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area chasing by cevians of a triangle - OMAPA Paraguay 2003.4
Triangle
A
B
C
ABC
A
BC
is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle
A
B
C
ABC
A
BC
. https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b495917b1ff.png
1
1
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numbers with product of digits 256 - OMAPA Paraguay 2003.1
How many numbers greater than
1.000
1.000
1.000
but less than
10.000
10.000
10.000
have as a product of their digits
256
256
256
?
5
1
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area chasing in a square - OMAPA Paraguay 2003.5
In a square
A
B
C
D
ABCD
A
BC
D
,
E
E
E
is the midpoint of side
B
C
BC
BC
. Line
A
E
AE
A
E
intersects line
D
C
DC
D
C
at
F
F
F
and diagonal
B
D
BD
B
D
at
G
G
G
. If the area
(
E
F
C
)
=
8
(EFC) = 8
(
EFC
)
=
8
, determine the area
(
G
B
E
)
(GBE)
(
GBE
)
.
3
1
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perfect square of 4 digits, 2 ages, 33 years later - OMAPA Paraguay 2003.3
Today the age of Pedro is written and then the age of Luisa, obtaining a number of four digits that is a perfect square. If the same is done in
33
33
33
years from now, there would be a perfect square of four digits . Find the current ages of Pedro and Luisa.
2
1
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3 digits wanted - OMAPA Paraguay 2003.2
With three different digits, all greater than
0
0
0
, six different three-digit numbers are formed. If we add these six numbers together the result is
4.218
4.218
4.218
. The sum of the three largest numbers minus the sum of the three smallest numbers equals
792
792
792
. Find the three digits.