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Chile Classification NMO Juniors
2001 Chile Classification NMO Juniors
2001 Chile Classification NMO Juniors
Part of
Chile Classification NMO Juniors
Subcontests
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2001 Chile Classification / Qualifying NMO Juniors XIII
p1. Juan was born before the year
2000
2000
2000
. On August
25
25
25
,
2001
2001
2001
he is as old as the sum of the digits of the year of his birth. Determine your date of birth and justify that it is the only possible solution. p2. Triangle
A
B
C
ABC
A
BC
is right isosceles. The figure below shows two basic ways to inscribe a square in it. The square
A
D
E
F
ADEF
A
D
EF
is known to have area
2250
2250
2250
. Determine the area of the square
G
H
I
J
GHIJ
G
H
I
J
. https://cdn.artofproblemsolving.com/attachments/9/3/84d0cda6e109aaf01fb2f3de4d1933f0f16f82.jpgp3. 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
3
3
3
, so that each pair of people is in exactly one of these groups. Also, show one of the possible formations of the groups. p4. Show that there are no positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
a
2
+
b
2
=
8
c
+
6
a^2 + b^2 = 8c + 6
a
2
+
b
2
=
8
c
+
6
. p5. In a circular roulette the numbers from
1
1
1
to
36
36
36
are randomly placed. Show that necessarily there must be
3
3
3
consecutive numbers whose sum is at least
55
55
55
. p6. Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in a circle of radius
r
r
r
, such that its diagonals are perpendicular at point
E
E
E
. Prove that
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
.
p7. In a rectangular board of
m
m
m
rows and
n
n
n
columns, place
1
1
1
or
0
0
0
in each of the
m
n
mn
mn
cells, so that the numbers of each row have the same sum
f
f
f
and those of each column have the same sum
c
c
c
. For a board with
m
=
15
m = 15
m
=
15
,
n
=
10
n = 10
n
=
10
, and for
f
=
4
f = 4
f
=
4
, determine the value of
c
c
c
that makes this assignment possible and show a way to do this assignment.PS. Juniors p2, p6, were posted also as [url=https://artofproblemsolving.com/community/c4h2691996p23368695]Seniors variation of p2, part of p6.