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Chile Junior Math Olympiad
2015 Chile Junior Math Olympiad
2015 Chile Junior Math Olympiad
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Chile Junior Math Olympiad
Subcontests
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2015 Chile NMO Juniors XXVII
[url=https://artofproblemsolving.com/community/c4h1845752p12426710]p1. On the plane, there is drawn a parallelogram
P
P
P
and a point
X
X
X
outside of
P
P
P
. Using only an ungraded rule, determine the point
W
W
W
that is symmetric to
X
X
X
with respect to the center
O
O
O
of
P
P
P
. [url=https://artofproblemsolving.com/community/c1068820h2927813p26188378]p2. Consider a triangle
△
A
B
C
\triangle ABC
△
A
BC
and a point
D
D
D
in segment
B
C
BC
BC
. The triangles
△
A
B
D
\triangle ABD
△
A
B
D
and
△
A
D
C
\triangle ADC
△
A
D
C
are similar in ratio
1
3
\frac{1}{\sqrt3}
3
1
. Determine the angles of the triangle
△
A
B
C
\triangle ABC
△
A
BC
. [url=https://artofproblemsolving.com/community/c6h2927811p26188370]p3. Consider a horizontal line
L
L
L
with
20
20
20
different points
P
1
,
P
2
,
.
.
.
,
P
20
P_1, P_2,..., P_{20}
P
1
,
P
2
,
...
,
P
20
on her. For each pair of points
P
i
P_i
P
i
,
P
j
P_j
P
j
a circle is drawn such that the segment
P
i
P
j
P_iP_j
P
i
P
j
is a diameter. Determine the maximum number of intersections between circles that can occur, considering only those that occur strictly above
L
L
L
. p4. The bottle in the figure has a circular base, and the bottom of it is a perfect cylinder. The upper part is not very well defined. With the aid of a graded ruler (with which you can measure distances), and a water tap, propose a method that allows you to estimate very precisely the total volume of the bottle. https://cdn.artofproblemsolving.com/attachments/6/d/cf341cb6892feb70fa51a4cf54c96738e53092.png [url=https://artofproblemsolving.com/community/c4h1845749p12426698]p5. A quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. Suppose that
∣
D
A
∣
=
∣
B
C
∣
=
2
|DA| =|BC|= 2
∣
D
A
∣
=
∣
BC
∣
=
2
and
∣
A
B
∣
=
4
|AB| = 4
∣
A
B
∣
=
4
. Let
E
E
E
be the point of intersection of lines
B
C
BC
BC
and
D
A
DA
D
A
. Suppose that
∠
A
E
B
=
6
0
o
\angle AEB = 60^o
∠
A
EB
=
6
0
o
and that
∣
C
D
∣
<
∣
A
B
∣
|CD| <|AB|
∣
C
D
∣
<
∣
A
B
∣
. Calculate the radius of the circle. p6. Determine all triples of positive integers
(
p
,
n
,
m
)
(p, n, m)
(
p
,
n
,
m
)
, with
p
p
p
a prime number, which satisfy the equation
p
m
−
n
3
=
27
p^m- n^3 = 27
p
m
−
n
3
=
27