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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1994 Chile National Olympiad
1994 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
7
1
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reflections of ray of light in rectangle mxn, billiard clone (Chile NMO 1994 P7)
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle of length
m
m
m
and width
n
n
n
, with
m
,
n
m, n
m
,
n
positive integers. Consider a ray of light that starts from
A
A
A
, reflects with an angle of
4
5
o
45^o
4
5
o
on an opposite side and continues reflecting away at the same angle.
∙
\bullet
∙
For any pair
(
m
,
n
)
(m,n)
(
m
,
n
)
, show that the ray meets a vertex at some point.
∙
\bullet
∙
Suppose
m
m
m
and
n
n
n
are coprime. Determine the number of reflections made by the ray of light before encountering a vertex for the first time.
5
1
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x^{1994} +1/x^{1994} if x +1/x=-1 (Chile NMO 1994 P5)
Let
x
x
x
be a number such that
x
+
1
x
=
−
1
x +\frac{1}{x}=-1
x
+
x
1
=
−
1
. Determine the value of
x
1994
+
1
x
1994
x^{1994} +\frac{1}{x^{1994}}
x
1994
+
x
1994
1
.
4
1
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max no of equal balls of radius 1 in 10x16x1 (Chile NMO 1994 P4)
Consider a box of dimensions
10
10
10
cm
×
16
\times 16
×
16
cm
×
1
\times 1
×
1
cm. Determine the maximum number of balls of diameter
1
1
1
cm that the box can contain.
3
1
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prime with n digits all equal to 1 (Chile NMO 1994 P3)
Let
x
x
x
be an integer of
n
n
n
digits, all equal to
1
1
1
. Show that if
x
x
x
is prime, then
n
n
n
is also prime.
1
1
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distance between 2 of 11 railway stations (Chile NMO 1994 P1)
A railway line is divided into ten sections by stations
E
1
,
E
2
,
.
.
.
,
E
11
E_1, E_2,..., E_{11}
E
1
,
E
2
,
...
,
E
11
. The distance between the first and the last station is
56
56
56
km. A trip through two consecutive stations never exceeds
12
12
12
km, and a trip through three consecutive stations is at least
17
17
17
Km. Calculate the distance between
E
2
E_2
E
2
and
E
7
E_7
E
7
.
6
1
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projection of any quarilateral can be a parallelogram (Chile 1994 P6)
On a sheet of transparent paper, draw a quadrilateral with Chinese ink, which is illuminated with a lamp. Show that it is always possible to locate the sheet in such a way that the shadow projected on the desk is a parallelogram.
2
1
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cutting a triangle to form a rectangle (Chile 1994 P2)
Show that it is possible to cut any triangle into several pieces, so that a rectangle is formed when they are joined together.