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Problems
Contests
National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2009 Paraguay Mathematical Olympiad
2009 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
5
1
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All possible values of the length of a segment
In a triangle
A
B
C
ABC
A
BC
, let
I
I
I
be its incenter. The distance from
I
I
I
to the segment
B
C
BC
BC
is
4
c
m
4 cm
4
c
m
and the distance from that point to vertex
B
B
B
is
12
c
m
12 cm
12
c
m
. Let
D
D
D
be a point in the plane region between segments
A
B
AB
A
B
and
B
C
BC
BC
such that
D
D
D
is the center of a circumference that is tangent to lines
A
B
AB
A
B
and
B
C
BC
BC
and passes through
I
I
I
. Find all possible values of the length
B
D
BD
B
D
.
4
1
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Sum of the terms of a sequence
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
be a sequence such that the arithmetic mean of the
n
n
n
terms is
n
n
n
. Consider
n
=
2009
n = 2009
n
=
2009
. Determine the sum of the
2009
2009
2009
terms of the sequence.
3
1
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Last digit is 1
Find out how many positive integers
n
n
n
not larger than
2009
2009
2009
exist such that the last digit of
n
20
n^{20}
n
20
is
1
1
1
.
2
1
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Measure of an angle
In a triangle
A
B
C
ABC
A
BC
(
∠
B
C
A
=
9
0
∘
\angle{BCA} = 90^{\circ}
∠
BC
A
=
9
0
∘
), let
D
D
D
be the intersection of
A
B
AB
A
B
with a circumference with diameter
B
C
BC
BC
. Let
F
F
F
be the intersection of
A
C
AC
A
C
with a line tangent to the circumference. If
∠
C
A
B
=
4
6
∘
\angle{CAB} = 46^{\circ}
∠
C
A
B
=
4
6
∘
, find the measure of
∠
C
F
D
\angle{CFD}
∠
CF
D
.
1
1
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Sum of an expression
Find the value of the following expression:
2
+
33
+
6
+
35
+
10
+
37
+
…
+
1194
+
629
+
1198
+
631
2 + 33 + 6 + 35 + 10 + 37 + \ldots + 1194 + 629 + 1198 + 631
2
+
33
+
6
+
35
+
10
+
37
+
…
+
1194
+
629
+
1198
+
631