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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1992 Chile National Olympiad
1992 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
3
1
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no of 1/2 in a sequence of fractions(Chile NMO 1992 P3)
Determine the number of times and the positions in which it appears
1
2
\frac12
2
1
in the following sequence of fractions:
1
1
,
2
1
,
1
2
,
3
1
,
2
2
,
1
3
,
4
1
,
3
2
,
2
3
,
1
4
,
.
.
.
,
1
1992
\frac11, \frac21, \frac12 , \frac31 , \frac22 , \frac13 , \frac41,\frac32,\frac23,\frac14,..., \frac{1}{1992}
1
1
,
1
2
,
2
1
,
1
3
,
2
2
,
3
1
,
1
4
,
2
3
,
3
2
,
4
1
,
...
,
1992
1
2
1
Hide problems
sum 1/P(C) , product of its elements (Chile NMO 1992 P2)
For a finite set of naturals
(
C
)
(C)
(
C
)
, the product of its elements is going to be noted
P
(
C
)
P(C)
P
(
C
)
. We are going to define
P
(
ϕ
)
=
1
P (\phi) = 1
P
(
ϕ
)
=
1
. Calculate the value of the expression
∑
C
⊆
{
1
,
2
,
.
.
.
,
n
}
1
P
(
C
)
\sum_{C \subseteq \{1,2,...,n\}} \frac{1}{P(C)}
C
⊆
{
1
,
2
,
...
,
n
}
∑
P
(
C
)
1
7
1
Hide problems
magic square nxn with fixed sum close to 1992 (Chile NMO 1992 P7)
∙
\bullet
∙
Determine a natural
n
n
n
such that the constant sum
S
S
S
of a magic square of
n
×
n
n \times n
n
×
n
(that is, the sum of its elements in any column, or the diagonal) differs as little as possible from
1992
1992
1992
.
∙
\bullet
∙
Construct or describe the construction of this magic square.
6
1
Hide problems
3 participants in M events in Mathlon (Chile NMO 1992 P6)
A Mathlon is a competition where there are
M
M
M
athletic events.
A
,
B
A, B
A
,
B
and
C
C
C
were the only participants of a Mathlon. In each event,
p
1
p_1
p
1
points were given to the first place,
p
2
p_2
p
2
points to the second place and
p
3
p_3
p
3
points to third place, with
p
1
>
p
2
>
p
3
>
0
p_1> p_2> p_3> 0
p
1
>
p
2
>
p
3
>
0
where
p
1
p_1
p
1
,
p
2
p_2
p
2
and
p
3
p_3
p
3
are integer numbers. The final result was
22
22
22
points for
A
A
A
,
9
9
9
for
B
B
B
, and
9
9
9
for
C
C
C
.
B
B
B
won the
100
100
100
meter dash. Determine
M
M
M
and who was the second in high jump.
1
1
Hide problems
2^n + 5 is a perfect square (Chile NMO 1992 P1)
Determine all naturals
n
n
n
such that
2
n
+
5
2^n + 5
2
n
+
5
is a perfect square.
5
1
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computational in a triangle (Chile 1992 L2 P5)
In the
△
A
B
C
\triangle ABC
△
A
BC
, points
M
,
I
,
H
M, I, H
M
,
I
,
H
are feet, respectively, of the median, bisector and height, drawn from
A
A
A
. It is known that
B
C
=
2
BC = 2
BC
=
2
,
M
I
=
2
−
3
MI = 2-\sqrt {3}
M
I
=
2
−
3
and
A
B
>
A
C
AB > AC
A
B
>
A
C
. a) Prove that
I
I
I
lies between
M
M
M
and
H
H
H
. b) Calculate
A
B
2
−
A
C
2
AB ^ 2-AC ^ 2
A
B
2
−
A
C
2
. c) Determine
A
B
A
C
\dfrac {AB} {AC}
A
C
A
B
. d) Find the measure of all the sides and angles of the triangle.
4
1
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equilateral with vertices on parallel lines (Chile 1992 L2 P4)
Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.