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El Salvador Contests
El Salvador Correspondence
2005 El Salvador Correspondence
2005 El Salvador Correspondence
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El Salvador Correspondence
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2005 El Salvador Correspondence / Qualifying NMO V
p1. For what values of
a
a
a
is
(
5
a
+
1
)
(
3
a
+
2
)
(5 a + 1)(3a + 2)
(
5
a
+
1
)
(
3
a
+
2
)
divisible by
15
15
15
? p2. Of the four-digit positive integers that are multiple of
9
9
9
, how many are there that have all their digits different from zero and different from each other? p3. A square with side
5
5
5
is divided into
25
25
25
unit squares by lines parallel to the sides. Let A be the set of the
16
16
16
interior points, which are vertices of the unit squares, but which are not on the sides of the initial square. What is the largest number of points in
A
A
A
that can be chosen so that any three of them not vertices of a right isosceles triangle? p4. In a parallelogram
A
B
C
D
ABCD
A
BC
D
,
B
C
=
2
A
B
BC = 2AB
BC
=
2
A
B
. The bisector of angle
B
B
B
intersects the extension of segment
C
D
CD
C
D
at
Q
Q
Q
and the bisector of angle
A
A
A
intersects segment
B
D
BD
B
D
at
M
M
M
and segment
B
C
BC
BC
at
P
P
P
. If
P
Q
=
6
PQ = 6
PQ
=
6
, find the length of the segment
M
B
MB
MB
. p5. Find the number of ways to write nonnegative integers in each box on a board of dimension
n
×
n
n \times n
n
×
n
, so that the sum of the numbers in each row and each column is equal to
3
3
3
and in each row and in each column there are only one or two numbers different from zero.