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El Salvador Correspondence
2018 El Salvador Correspondence
2018 El Salvador Correspondence
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El Salvador Correspondence
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2018 El Salvador Correspondence / Qualifying NMO XVIII
p1. Manuel writes a list with the positive integers from
1
1
1
to
2018
2018
2018
. Next, he places a sign negative in front of all powers of
2
2
2
and a positive sign in front of all other numbers, that is to say:
−
1
−
2
+
3
−
4
+
5
+
6
+
7
−
8
+
.
.
.
+
2018
-1 - 2 + 3 - 4 + 5 + 6 + 7 - 8 +... + 2018
−
1
−
2
+
3
−
4
+
5
+
6
+
7
−
8
+
...
+
2018
Calculate the result that Manuel obtains when carrying out the indicated operations. p2. Determine the number of whole numbers
n
n
n
between
1
1
1
and
2018
2018
2018
, inclusive, such that the product
(
1
+
1
2
)
(
1
+
1
3
)
.
.
.
(
1
+
1
n
)
\left(1+\frac12\right)\left(1+\frac13\right)...\left(1+\frac{1}{n}\right)
(
1
+
2
1
)
(
1
+
3
1
)
...
(
1
+
n
1
)
is also an integer. p3. Determine all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers that satisfy the equation
(
x
+
y
)
2
=
(
x
+
2018
)
(
y
−
2018
)
(x + y)^2 = (x + 2018) (y - 2018)
(
x
+
y
)
2
=
(
x
+
2018
)
(
y
−
2018
)
p4. In triangle
A
B
C
ABC
A
BC
, the angle
∠
A
C
B
=
9
0
o
\angle ACB =90^o
∠
A
CB
=
9
0
o
. The internal bisectors of the angles
∠
B
A
C
\angle BAC
∠
B
A
C
and
∠
A
B
C
\angle ABC
∠
A
BC
intersect the sides
B
C
BC
BC
and
C
A
CA
C
A
at
P
P
P
and
Q
Q
Q
, respectively. Points
M
M
M
and
N
N
N
are the feet of the perpendiculars from
P
P
P
and
Q
Q
Q
to side
A
B
AB
A
B
. Calculate the measure of the angle
∠
M
C
N
\angle MCN
∠
MCN
. p5. Carlos takes turns with Rodrigo on the next
5
×
6
5 \times 6
5
×
6
board. https://cdn.artofproblemsolving.com/attachments/8/e/388ee57b67289ac9b50565d1444ba5d3e1d958.png A move consists of placing a piece of one of the following three types on the board: https://cdn.artofproblemsolving.com/attachments/b/4/aa96fcb2b5acde281ca0b8b7fa1e8bd00e0083.png Parts can be rotated before being located and not allowed to overlap with those in place previously on the board. Whoever finishes covering the board wins. If Carlos moves first, determine the player who can secure the victory and the strategy that must continue to win.