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National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2013 El Salvador Correspondence
2013 El Salvador Correspondence
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El Salvador Correspondence
Subcontests
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2013 El Salvador Correspondence / Qualifying NMO XIII
p1. Determine the unit digit of the number resulting from the following sum
201
3
1
+
201
3
2
+
201
3
3
+
.
.
.
+
201
3
2012
+
201
3
2013
2013^1 + 2013^2 + 2013^3 + ... + 2013^{2012} + 2013^{2013}
201
3
1
+
201
3
2
+
201
3
3
+
...
+
201
3
2012
+
201
3
2013
2. Every real number a can be uniquely written as
a
=
[
a
]
+
{
a
}
a = [a] +\{a\}
a
=
[
a
]
+
{
a
}
, where
[
a
]
[a]
[
a
]
is an integer and
0
≤
{
a
}
<
1
0\le \{a\}<1
0
≤
{
a
}
<
1
. For example, if
a
=
2.12
a = 2.12
a
=
2.12
, then
[
2.12
]
=
2
[2.12] = 2
[
2.12
]
=
2
and
{
2.12
}
=
0.12
\{2.12\} = 0.12
{
2.12
}
=
0.12
. Given the:
x
+
[
y
]
+
{
z
}
=
4.2
x + [y] + \{z\} = 4.2
x
+
[
y
]
+
{
z
}
=
4.2
y
+
[
z
]
+
{
x
}
=
3.6
y + [z] + \{x\} = 3.6
y
+
[
z
]
+
{
x
}
=
3.6
z
+
[
x
]
+
{
y
}
=
2.0
z + [x] + \{y\} = 2.0
z
+
[
x
]
+
{
y
}
=
2.0
Determine the value of
x
−
y
+
z
x - y + z
x
−
y
+
z
. p3. Determine all pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
2
(
x
+
y
)
+
x
y
=
x
2
+
y
2
2 (x + y) + xy = x^2 + y^2
2
(
x
+
y
)
+
x
y
=
x
2
+
y
2
. p4. Consider the following arrangement of dots on the board in the figure. Determine the number of ways three of these points can be selected, to be the vertices of a right triangle whose legs are parallel to the sides of the board. https://cdn.artofproblemsolving.com/attachments/3/1/a9025e6e6f41f6a8d745a1c695b61640e9c691.png p5. In an acute triangle
A
B
C
ABC
A
BC
,
∠
A
=
3
0
o
\angle A=30^o
∠
A
=
3
0
o
. Let
D
D
D
and
E
E
E
be the feet of the altitudes drawn from
B
B
B
and
C
C
C
, respectively. Let
F
F
F
and
G
G
G
be the midpoints of sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. Prove that segments
D
G
DG
D
G
and
E
F
EF
EF
are perpendicular.