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National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2002 Paraguay Mathematical Olympiad
2002 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
2
1
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volume of rectangular parallelepiped OMAPA Paraguay 2002.2
In the rectangular parallelepiped in the figure, the lengths of the segments
E
H
EH
E
H
,
H
G
HG
H
G
, and
E
G
EG
EG
are consecutive integers. The height of the parallelepiped is
12
12
12
. Find the volume of the parallelepiped. https://cdn.artofproblemsolving.com/attachments/6/4/f74e7fed38c815bff5539613f76b0c4ca9171b.png
5
1
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computational in right trapezoid - OMAPA Paraguay 2002.5
In a trapezoid
A
B
C
D
ABCD
A
BC
D
, the side
D
A
DA
D
A
is perpendicular to the bases
A
B
AB
A
B
and
C
D
CD
C
D
. Also
A
B
=
45
AB=45
A
B
=
45
,
C
D
=
20
CD =20
C
D
=
20
,
B
C
=
65
BC =65
BC
=
65
. Let
P
P
P
be a point on the side
B
C
BC
BC
such that
B
P
=
45
BP=45
BP
=
45
and let
M
M
M
be the midpoint of
D
A
DA
D
A
. Calculate the length of
P
M
PM
PM
.
4
1
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n + 195 and n -274 are perfect cubes OMAPA Paraguay 2002.4
Find all natural numbers
n
n
n
for which
n
+
195
n + 195
n
+
195
and
n
−
274
n - 274
n
−
274
are perfect cubes.
3
1
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3 3-digits numbers wanted OMAPA Paraguay 2002.3
With three different digits, six-digit numbers are written, multiples of
3
3
3
. One of the the digits are in the unit's place, another in the hundred's place, and the third in the remaining places. If we take out two units from the hundred's digit and add these to the unit's digit, the number is left with all the same digits. Find the numbers.
1
1
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max no of students for a single dentist OMAPA Paraguay 2002.1
There are
12
12
12
dentists in a clinic near a school. The students of the
5
5
5
th year, who are
29
29
29
, attend the clinic. Each dentist serves at least
2
2
2
students. Determine the greater number of students that can attend to a single dentist .