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National and Regional Contests
Chile Contests
Chile Classification NMO
2001 Chile Classification NMO Seniors
2001 Chile Classification NMO Seniors
Part of
Chile Classification NMO
Subcontests
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2001 Chile Classification / Qualifying NMO Seniors XIII
p1. All positive fractions less than one are considered, whose denominator is
2001
2001
2001
and whose numerator is a number that has no common divisors with
2001
2001
2001
. Calculate the sum of these fractions. p2. Triangle
A
B
C
ABC
A
BC
is right isosceles. The figure below shows two basic ways to inscribe a square in it. Prove that the square
A
D
E
F
ADEF
A
D
EF
has a greater area than the square
G
H
I
J
GHIJ
G
H
I
J
. https://cdn.artofproblemsolving.com/attachments/9/3/84d0cda6e109aaf01fb2f3de4d1933f0f16f82.jpg p3. Given
9
9
9
people, show that there exists a value of
n
n
n
such that with people you can form
n
n
n
groups of a
3
3
3
, so that each pair of people is in exactly one of these groups and show a corresponding conformation of these groups. If the same number of groups must be formed, but of
6
6
6
people each and with the condition that each pair is in exactly
k
k
k
groups, determine if there is a value of
k
k
k
that makes it possible for the problem to have a solution and, if affirmative, display a corresponding formation. p4. Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. Side
A
B
AB
A
B
is extended to a point
E
E
E
, such that
B
E
=
B
C
BE = BC
BE
=
BC
and the side
A
D
AD
A
D
is extended to a point
F
F
F
, such that
D
F
=
D
C
DF = DC
D
F
=
D
C
.
∙
\bullet
∙
Prove that points
E
E
E
,
C
C
C
, and
F
F
F
are collinear.
∙
\bullet
∙
Prove that the perpendicular to line
A
E
AE
A
E
at point
E
E
E
, the perpendicular to line
A
F
AF
A
F
at point
F
F
F
, the bisector of the angle
E
A
F
EAF
E
A
F
and the perpendicular to the diagonal
B
D
BD
B
D
by the vertex
C
C
C
are all concurrent at a point
G
G
G
. https://cdn.artofproblemsolving.com/attachments/c/1/e0d585f30c4c9ca274e1ce1599128e3e21a08c.png p5. Consider two positive integers
x
x
x
and
y
y
y
that satisfy the relation
3
x
2
+
x
=
4
y
2
+
y
3x^2 + x = 4y^2 + y
3
x
2
+
x
=
4
y
2
+
y
. Prove that the numbers
(
x
−
y
)
(x-y)
(
x
−
y
)
,
(
3
x
+
3
y
+
1
)
(3x + 3y + 1)
(
3
x
+
3
y
+
1
)
,
(
4
x
+
4
y
+
1
)
(4x + 4y + 1)
(
4
x
+
4
y
+
1
)
are three perfect squares. p6. Let
A
B
C
D
ABCD
A
BC
D
be an quadrilateral inscribed in a circle of radius r, and let E be the point where its diagonals intersect.
∙
\bullet
∙
Prove if the diagonals are perpendicular to each other, then
A
E
2
+
B
E
2
+
C
E
2
+
D
E
2
=
4
r
2
AE^2 + BE^2 + CE^2 + DE^2 = 4r^2
A
E
2
+
B
E
2
+
C
E
2
+
D
E
2
=
4
r
2
.
∙
\bullet
∙
If the previous relation is fulfilled, are the diagonals of the quadrilateral necessarily perpendicular? p7 On a rectangular board with
m
m
m
rows and n columns, place in each of the squares
m
n
mn
mn
a
1
1
1
or a
0
0
0
, so that the numbers in each add the same amount
f
f
f
and those in each column, the same amount
c
c
c
. Show that a necessary and sufficient condition for this assignment to be possible is that
m
f
=
n
c
mf = nc
m
f
=
n
c
.PS. Seniors p2, part of p6, were posted also as [url=https://artofproblemsolving.com/community/c4h2689068p23335606]Juniors variation p2, p6.