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Chile Junior Math Olympiad
1999 Chile Junior Math Olympiad
1999 Chile Junior Math Olympiad
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Chile Junior Math Olympiad
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1999 Chile NMO Juniors XI
[url=https://artofproblemsolving.com/community/c1068820h2917815p26064153]p1. In a triangle
A
B
C
ABC
A
BC
the median
C
M
CM
CM
is such that
∠
A
C
B
\angle ACB
∠
A
CB
is divided in the ratio
1
:
2
1: 2
1
:
2
. The median
C
M
CM
CM
is extended to point
D
D
D
such that
∠
D
A
C
=
9
0
o
\angle DAC = 90^o
∠
D
A
C
=
9
0
o
. Show that
C
D
=
2
B
C
CD = 2BC
C
D
=
2
BC
. [url=https://artofproblemsolving.com/community/c1068820h2917816p26064158]p2. Given two tangent equal circles with centers
O
O
O
and
O
1
O_1
O
1
respectively . Determine the distance at which a parallel line
O
O
1
OO_1
O
O
1
should be drawn so that the hatched areas in the figure are equal. https://cdn.artofproblemsolving.com/attachments/0/3/96182b4e3515734d7c5531b0e1a8e80b70ea16.png p3. There are
1999
1999
1999
balls numbered from
1
1
1
to
1999
1999
1999
. Find all positive integers
k
k
k
with the following property: The total of balls can be separated into
k
k
k
groups, so that the sum of the numbers of the balls in each of the groups is the same. p4. Three married couples, the Alvarezes, the Barros and the Castros, went to buy Christmas gifts. Each woman bought a quantity
p
p
p
of gifts in
1000
p
2
1000p^2
1000
p
2
pesos and each woman (Miriam, Francisca and Gladys) spent
147
,
000
147,000
147
,
000
pesos more than her respective husband. Miriam is known to have bought
9
9
9
fewer gifts that Mr. Barros and that Francisca bought
47
47
47
less gifts than Mr. Castro. Who is married with Gladys? [url=https://artofproblemsolving.com/community/c1068820h2917817p26064161]p5. In the figure,
A
D
AD
A
D
and
C
B
CB
CB
are two chords of a circle, which intersect at
E
E
E
, and
F
G
FG
FG
is the bisector of
∠
A
E
D
\angle AED
∠
A
E
D
. Show that
A
F
⋅
B
G
=
C
G
⋅
F
D
AF \cdot BG = CG \cdot FD
A
F
⋅
BG
=
CG
⋅
F
D
. https://cdn.artofproblemsolving.com/attachments/6/0/0e29eeaae46afea0f8c646ec223a27e1134e6a.png p6. Does the equation
x
2
+
y
2
=
1999
x
y
x^2 + y^2 = 1999xy
x
2
+
y
2
=
1999
x
y
has positive integer solutions? [url=https://artofproblemsolving.com/community/c1068820h2946119p26374834]p7. In a circle,
140
140
140
points are located. Show that, among them, you can choose
1999
1999
1999
pairs so that the arc subtended by the two points of each pair is less than or equal to
12
0
o
120^o
12
0
o
.