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National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2010 El Salvador Correspondence
2010 El Salvador Correspondence
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El Salvador Correspondence
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2010 El Salvador Correspondence / Qualifying NMO X
p1. In the rectangle
A
B
C
D
ABCD
A
BC
D
there exists a point
P
P
P
on the side
A
B
AB
A
B
such that
∠
P
D
A
=
∠
B
D
P
=
∠
C
D
B
\angle PDA = \angle BDP = \angle CDB
∠
P
D
A
=
∠
B
D
P
=
∠
C
D
B
and
D
A
=
2
DA = 2
D
A
=
2
. Find the perimeter of the triangle
P
B
D
PBD
PB
D
. p2. Write in each of the empty boxes of the following pyramid a number natural greater than
1
1
1
, so that the number written in each box is equal to the product of the numbers written in the two boxes on which it is supported. https://cdn.artofproblemsolving.com/attachments/3/f/395e1c09fd955ed7ba6f9fea23ef68a00a4880.png p3. In how many ways can seven numbers be chosen from
1
1
1
to
9
9
9
such that their sum is a multiple of
3
3
3
? p4. Simplify the product
p
=
(
10
+
1
)
(
1
0
2
+
1
)
(
(
1
0
2
)
2
+
1
)
(
(
1
0
2
)
4
+
1
)
(
(
1
0
2
)
8
+
1
)
.
.
.
(
(
1
0
2
)
1024
+
1
)
p = (10 + 1) (10^2 + 1) ((10^2)^2 + 1) ((10^2)^4 + 1) ((10^2)^8 + 1)...((10^2)^{1024} +1)
p
=
(
10
+
1
)
(
1
0
2
+
1
)
((
1
0
2
)
2
+
1
)
((
1
0
2
)
4
+
1
)
((
1
0
2
)
8
+
1
)
...
((
1
0
2
)
1024
+
1
)
p5. A lattice point is a point on the Cartesian plane with integer coordinates. We randomly select five lattice points. Show that there are two of them that form a line segment whose midpoint is also a lattice point.