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Chile Junior Math Olympiad
2008 Chile Junior Math Olympiad
2008 Chile Junior Math Olympiad
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Chile Junior Math Olympiad
Subcontests
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2008 Chile NMO Juniors XX
[url=https://artofproblemsolving.com/community/c1068820h2935441p26269039]p1. Alberto wants to invite Ximena to do math, instead of pointing out her exactly which are the Transantiago uses that serve him, he says: ''The numbers that lead to my house have three digits, the digit on the left being not null. Also, these numbers are multiples of
13
13
13
, and the second digit of them is the average of the other two. '' What are the bus lines that lead to Alberto's house? [url=https://artofproblemsolving.com/community/c4h1845746p12426674]p2. In a circle of radius
1
1
1
a diameter
P
Q
PQ
PQ
is drawn and an equilateral triangle with base
A
B
AB
A
B
parallel to
P
Q
PQ
PQ
is inscribed. The segment
P
Q
PQ
PQ
cuts to the side
B
C
BC
BC
at the point
R
R
R
. Is the length
P
R
PR
PR
smaller, equal, or greater than the length of a quarter of the circumference? https://cdn.artofproblemsolving.com/attachments/a/d/f11bdbd06f011fc6cf474fc566d941b9370950.png p3. We say that a quadruple of positive integers is primitive if at least two of its elements they are coprime (that is, they have no common factors). i) Find all the primitive quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
such that
a
b
=
c
d
\frac{a}{b}=\frac{c}{d}
b
a
=
d
c
and
a
+
b
+
c
=
d
.
a + b + c = d .
a
+
b
+
c
=
d
.
ii). Prove that for all primitive quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
satisfying these two properties, also satisfy
1
a
=
1
b
+
1
c
+
1
d
\frac{1}{a}=\frac{1}{b}+\frac{1}{c}+\frac{1}{d}
a
1
=
b
1
+
c
1
+
d
1
[url=https://artofproblemsolving.com/community/c4h2917776p26063733]p4. Let
C
C
C
be a set of
n
>
3
n> 3
n
>
3
points in the plane such that, given any three points in
C
C
C
, there is a fourth point in
C
C
C
so these four points are the vertices of a square. What are the possible values of
n
n
n
? p5. There are three real numbers such that
a
3
+
b
3
+
c
3
=
3
a
b
c
a^3 + b^3 + c^3 = 3abc
a
3
+
b
3
+
c
3
=
3
ab
c
. If the value of
a
+
b
+
c
a + b + c
a
+
b
+
c
is not zero, prove that
a
=
b
=
c
a = b = c
a
=
b
=
c
. [url=https://artofproblemsolving.com/community/c1068820h2935449p26269088]p6. When planning a trip from Temuco to the extreme north of the country, a truck driver notices that to cross the Atacama desert you must cross a distance of
800
800
800
km between two stations consecutive service. Your truck can only store
50
50
50
liters of benzene, and has a yield of
10
10
10
km per liter. The trucker can leave gasoline stored in cans on the side of the road in different points along the way. For example, with an initial total charge of
50
50
50
liters you can travel
100
100
100
km, leave
30
30
30
liters stored at the point you reached, and return to the starting point (with zero load) to refuel. The trucker decides to start the trip and arrives at the first service station with a zero load of fuel.a) Can the trucker cross the desert if at this service station the total supply is
140
140
140
liters? b) Can the trucker cross the desert if the total supply of gasoline at the station is
180
180
180
liters?PS. Problems 1 and 6 were also proposed as Seniors P1 and P5 respectively.