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National and Regional Contests
Chile Contests
Chile Classification NMO
1995 Chile Classification NMO
1995 Chile Classification NMO
Part of
Chile Classification NMO
Subcontests
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1995 Chile Classification / Qualifying NMO VII
p1. Let
a
a
a
be a positive integer. Prove that the equation
x
2
−
y
2
=
a
3
x^2-y^2 = a^3
x
2
−
y
2
=
a
3
always has solutions
x
,
y
x, y
x
,
y
in the integers. p2. A reporter arrived or after a
100
100
100
-meter dash ended. To write his article, got the following information from five people who saw the arrival of the race:
∙
\bullet
∙
Alejandro: Juan came second and Oscar came third.
∙
\bullet
∙
Boris: Pedro came third and Tom came fifth.
∙
\bullet
∙
Cristian: Tom thus came first and Pedro came second.
∙
\bullet
∙
Jorge: Juan came second and Raul came fourth.
∙
\bullet
∙
Diego: Oscar came first and Raul came fourth. It was clear to the reporter that he did not have the correct information, so he consulted with a colleague, but he only mentioned that in each answer there was a correct and an incorrect position. Help the reporter to find the correct order of arrival. p3.
A
B
C
D
E
F
G
H
ABCDEFGH
A
BC
D
EFG
H
is a cube with side
2
2
2
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
N
N
N
the midpoint of
E
F
EF
EF
. Find the area of the quadrilateral
A
M
H
N
AMHN
A
M
H
N
. p4. Given a trapezoid
A
B
C
D
ABCD
A
BC
D
, with
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
and
A
D
=
D
C
=
A
B
2
AD = DC =\frac{AB}{2}
A
D
=
D
C
=
2
A
B
.Determine
∠
A
C
B
\angle ACB
∠
A
CB
. p5. Are there two positive integers
a
,
b
a, b
a
,
b
whose sum is
1995
1995
1995
and whose product is a multiple of nineteen ninety five? p6. A rectangle
A
B
C
D
ABCD
A
BC
D
is divided into
5
5
5
rectangles
R
1
,
R
2
,
R
3
,
R
4
R_1, R_2, R_3, R_4
R
1
,
R
2
,
R
3
,
R
4
and
R
5
R_5
R
5
, as indicated in the figure. It is known that
R
5
R_5
R
5
is a square and that the areas of
R
1
,
R
2
,
R
3
R_1, R_2, R_3
R
1
,
R
2
,
R
3
and
R
4
R_4
R
4
are equal to each other. Prove that
A
B
C
D
ABCD
A
BC
D
is a square. ([color=#f00]missing figure) p7. Suppose that in a room no boy dances with all the girls, but that each girl dances with at least one boy. Prove that there are two boys:
m
m
m
and
m
′
m'
m
′
, and two girls,
n
n
n
and
n
′
n'
n
′
such that
m
m
m
dances with
n
n
n
, and
m
′
m'
m
′
dances with
n
′
n'
n
′
, but
m
m
m
does not dance with
n
′
n'
n
′
, and
m
′
m'
m
′
does not dance with
n
n
n
.