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Contests
National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2004 Paraguay Mathematical Olympiad
2004 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
1
1
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staircase by 2004 bricks - OMAPA Paraguay 2004.1
Stairs are built by laying bricks as shown in the figure. If you have
2004
2004
2004
bricks to build a staircase: a) How many steps (= escalones) will the ladder have? b) How many bricks will there be left over? https://cdn.artofproblemsolving.com/attachments/4/b/c1b80b374daeda4e33e1bb45be1d11f4b89590.png
5
1
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A^2 is a 4digit number, with 5 in ten's digit - OMAPA Paraguay 2004.5
We have an integer
A
A
A
such that
A
2
A^2
A
2
is a four digit number, with
5
5
5
in the ten's place . Find all possible values of
A
A
A
.
4
1
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AF, AE trisect diagonal BD of a square ABCD - OMAPA Paraguay 2004.4
In a square
A
B
C
D
ABCD
A
BC
D
,
E
E
E
is the midpoint of
B
C
BC
BC
and
F
F
F
is the midpoint of
C
D
CD
C
D
. Prove that
A
F
AF
A
F
and
A
E
AE
A
E
divide the diagonal
B
D
BD
B
D
in three equal segments.
3
1
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computational, perpendiculars, equilateral - OMAPA Paraguay 2004.3
In an equilateral triangle
A
B
C
ABC
A
BC
, whose side is
4
4
4
, the line perpendicular to
A
B
AB
A
B
is drawn through the point
A
A
A
, the line perpendicular to
B
C
BC
BC
through point
B
B
B
and the line perpendicular to
C
A
CA
C
A
through point
C
C
C
. These three lines determine another triangle. Calculate the perimeter of this triangle
2
1
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2x + 2,x + 4,x + 2 sides of right triangle - OMAPA Paraguay 2004.2
Determine for what values of
x
x
x
the expressions
2
x
+
2
2x + 2
2
x
+
2
,
x
+
4
x + 4
x
+
4
,
x
+
2
x + 2
x
+
2
can represent the sidelengths of a right triangle.