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Problems
Contests
National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2011 Paraguay Mathematical Olympiad
2011 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
5
1
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Incenter and geocenter parallel to a cathetus
In a rectangle triangle, let
I
I
I
be its incenter and
G
G
G
its geocenter. If
I
G
IG
I
G
is parallel to one of the catheti and measures
10
c
m
10 cm
10
c
m
, find the lengths of the two catheti of the triangle.
4
1
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Division in parts inversely proportional to some numbers
A positive integer
N
N
N
is divided in
n
n
n
parts inversely proportional to the numbers
2
,
6
,
12
,
20
,
…
2, 6, 12, 20, \ldots
2
,
6
,
12
,
20
,
…
The smallest part is equal to
1
400
N
\frac{1}{400} N
400
1
N
. Find the value of
n
n
n
.
3
1
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Division with specific quotients and remainders
If number
a
a
a
a
‾
\overline{aaaa}
aaaa
is divided by
b
b
‾
\overline{bb}
bb
, the quotient is a number between
140
140
140
and
160
160
160
inclusively, and the remainder is equal to
(
a
−
b
)
(
a
−
b
)
‾
\overline{(a-b)(a-b)}
(
a
−
b
)
(
a
−
b
)
. Find all pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
that satisfy this.
2
1
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Distance of midpoints
In a triangle
A
B
C
ABC
A
BC
, let
D
D
D
and
E
E
E
be the midpoints of
A
C
AC
A
C
and
B
C
BC
BC
respectively. The distance from the midpoint of
B
D
BD
B
D
to the midpoint of
A
E
AE
A
E
is
4.5
4.5
4.5
. What is the length of side
A
B
AB
A
B
?
1
1
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Sum of an expression
Find the value of the following expression:
1
2
+
(
1
3
+
2
3
)
+
(
1
4
+
2
4
+
3
4
)
+
…
+
(
1
1000
+
2
1000
+
…
+
999
1000
)
\frac{1}{2} + (\frac{1}{3} + \frac{2}{3}) + (\frac{1}{4} + \frac{2}{4} + \frac{3}{4}) + \ldots + (\frac{1}{1000} + \frac{2}{1000} + \ldots + \frac{999}{1000})
2
1
+
(
3
1
+
3
2
)
+
(
4
1
+
4
2
+
4
3
)
+
…
+
(
1000
1
+
1000
2
+
…
+
1000
999
)