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2000 Chile Junior Math Olympiad
2000 Chile Junior Math Olympiad
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2000 Chile NMO Juniors XII
p1. Professor David had a serious accident when he went to the maternity ward where his wife give birth to a baby. Due to the accident he died, but before he reached to say his last will: If a girl is born, she must inherit
2
/
3
2/3
2/3
of his fortune, leaving then
1
/
3
1/3
1/3
for the widow, but if a boy is born, he shall inherit only
1
/
4
1/4
1/4
of fortune, leaving then
3
/
4
3/4
3/4
for the widow. It turns out that the lady had twins, a boy and a girl. How should the inheritance be distributed so that the teacher's will is fulfilled? [url=https://artofproblemsolving.com/community/c4h2917444p26058924]p2. Prove that with two circles of radius
r
<
1
r <1
r
<
1
it is not possible to cover a circle of radius
1
1
1
. p3. The balls numbered from
1
1
1
to
2000
2000
2000
are placed in one row (not necessarily in order numerical), and it is observed that for any ball the difference between the position it occupies in the row and the number with which it is numbered is the same for all the balls. Calculate the possible possible values of said difference. p4. Using
21
21
21
tiles, some of which are white and others are black, a rectangle is filled by
3
×
7
3\times 7
3
×
7
tiles. Show that there is always a rectangle within it such that its vertices have the same color. [url=https://artofproblemsolving.com/community/c4h1846801p12438187]p5. A point
P
P
P
interior to a
△
A
B
C
\vartriangle ABC
△
A
BC
satisfies:
∠
P
B
A
=
∠
P
C
A
=
1
3
(
∠
A
B
C
+
∠
A
C
B
)
\angle PBA = \angle PCA = \dfrac {1} {3} (\angle ABC + \angle ACB)
∠
PB
A
=
∠
PC
A
=
3
1
(
∠
A
BC
+
∠
A
CB
)
. Prove that
A
C
+
P
B
A
B
=
A
B
+
P
C
A
C
\dfrac {AC + PB} {AB} = \dfrac {AB + PC} {AC}
A
B
A
C
+
PB
=
A
C
A
B
+
PC
p6. The number
518
518
518
has the curious property that when taking the average of the
6
6
6
numbers obtained by reordering their digits the same number is obtained. Find all the numbers of three digits that have also this property. [url=https://artofproblemsolving.com/community/c1068820h2946121p26374845]p7. Claudio, who loves to play, challenges Andr to the following game:
n
n
n
points are scored on a paper. A move consists of joining two points by a line and placing a new point on the line. At most, three strokes can reach each point. A point can join itself by counting that
2
2
2
segments arrive. The player who cannot make a move on his turn loses. Andr accepts the challenge, but he has a doubt if they will ever finish playing.
∙
\bullet
∙
Prove that the game ends at
∙
\bullet
∙
Prove that for
n
=
2000
n = 2000
n
=
2000
, the one who starts first has a winning strategy.