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Contests
National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2006 Paraguay Mathematical Olympiad
2006 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
5
1
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Problem 5
Let
A
B
C
ABC
A
BC
be a triangle, and let
P
P
P
be a point on side
B
C
BC
BC
such that
B
P
P
C
=
1
2
\frac{BP}{PC}=\frac{1}{2}
PC
BP
=
2
1
. If
∡
A
B
C
\measuredangle ABC
∡
A
BC
=
=
=
4
5
∘
45^{\circ}
4
5
∘
and
∡
A
P
C
\measuredangle APC
∡
A
PC
=
=
=
6
0
∘
60^{\circ}
6
0
∘
, determine
∡
A
C
B
\measuredangle ACB
∡
A
CB
without trigonometry.
4
1
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Problem 4
Consider all pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
, with
a
<
b
a<b
a
<
b
, such that
a
+
b
=
2
,
160
\sqrt{a} +\sqrt{b} = \sqrt{2,160}
a
+
b
=
2
,
160
Determine all possible values of
a
a
a
.
3
1
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Problem 3
Let
Γ
A
\Gamma_A
Γ
A
,
Γ
B
\Gamma_B
Γ
B
,
Γ
C
\Gamma_C
Γ
C
be circles such that
Γ
A
\Gamma_A
Γ
A
is tangent to
Γ
B
\Gamma_B
Γ
B
and
Γ
B
\Gamma_B
Γ
B
is tangent to
Γ
C
\Gamma_C
Γ
C
. All three circles are tangent to lines
L
L
L
and
M
M
M
, with
A
A
A
,
B
B
B
,
C
C
C
being the tangency points of
M
M
M
with
Γ
A
\Gamma_A
Γ
A
,
Γ
B
\Gamma_B
Γ
B
,
Γ
C
\Gamma_C
Γ
C
, respectively. Given that
12
=
r
A
<
r
B
<
r
C
=
75
12=r_A<r_B<r_C=75
12
=
r
A
<
r
B
<
r
C
=
75
, calculate:a) the length of
r
B
r_B
r
B
. b) the distance between point
A
A
A
and the point of intersection of lines
L
L
L
and
M
M
M
.
2
1
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Problem 2
Consider all right triangles with integer sides such that the length of the hypotenuse and one of the two sides are consecutive. How many such triangles exist?
1
1
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Problem 1
What are the last two digits of the decimal representation of
2
1
2006
21^{2006}
2
1
2006
?