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Chile Classification NMO
1998 Chile Classification NMO
1998 Chile Classification NMO
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Chile Classification NMO
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1998 Chile Classification / Qualifying NMO X
p1. From each natural number
n
n
n
less than
100
100
100
, two digits are subtracted from the sum of the squares. For what values of
n
n
n
is this difference maximum? p2. In a triangle
A
B
C
ABC
A
BC
, isosceles and right in
B
B
B
, a point
D
D
D
is taken on the hypotenuse and is projected perpendicularly at points
E
E
E
and
F
F
F
of legs
A
B
AB
A
B
and
B
C
BC
BC
, respectively, dividing thus the triangle
A
B
C
ABC
A
BC
in the triangles
A
E
D
AED
A
E
D
,
D
F
C
DFC
D
FC
and the rectangle
E
B
F
D
EBFD
EBF
D
. If each leg measures
a
a
a
, show that at least one of the areas of these three figures of this division is greater than or equal to
2
9
a
2
\frac29 a^2
9
2
a
2
. p3. A normal year has
365
365
365
days. A leap year has
366
366
366
days, a leap year being a year not divisible by
4
4
4
, except those that being divisible by
10
10
10
are not divisible by
400
400
400
(for example 1900 was not leap, but the year
2000
2000
2000
will be such). Remembering that today is Saturday August
22
22
22
,
1998
1998
1998
, what day of the week was September
18
18
18
,
1810
1810
1810
? p4. Miriam invited nine boys and eight girls for her birthday. Her mother prepared T-shirts with the numbers from
1
1
1
to
18
18
18
and distributed them to all the participants of this. During a dance, the mother observed that the sum of the numbers of each pair was a perfect square. How were the nine couples made up? p5. Let
H
H
H
be the point where the altitudes of a triangle
A
B
C
ABC
A
BC
intersect. Show that the angle formed by the radii of the circumferences of diameter
A
H
AH
A
H
and
B
C
BC
BC
at their points of intersection, is a right angle. p6.In how many ways can
10
10
10
people be photographed, sitting in a row, so that four of them
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
always remain in the same relative order, that is,
a
a
a
is always at the left of
b
b
b
, that
b
b
b
is always to the left of
c
c
c
and that
c
c
c
is always to the left of
d
d
d
? p7. Find a number that is divisible by
1998
1998
1998
and such that the sum of its digits when written in the decimal system it is equal to
1998
1998
1998
.