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El Salvador Correspondence
2019 El Salvador Correspondence
2019 El Salvador Correspondence
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El Salvador Correspondence
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2019 El Salvador Correspondence / Qualifying NMO XIX
p1. In an infinite grid in all directions, each square is colored black or white. Each second the color of the squares changes as follows: if a white and a black square are adjacent, in the next second they both change color. It is known that at the beginning there is a single white box on the entire grid. Determine the number of white cells after
2019
2019
2019
seconds. p2. In the figure,
A
B
C
D
ABCD
A
BC
D
is a square with side
1
1
1
. Performing a rotation of
6
0
o
60^o
6
0
o
with center at
A
A
A
, obtain a second square
A
P
Q
R
APQR
A
PQR
. Determine the area of the pentagon
A
B
C
Q
R
ABCQR
A
BCQR
. https://cdn.artofproblemsolving.com/attachments/1/c/aafb52c860bc2f4e6f540afbfbff4d08e35eeb.png p3. Determine all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of nonzero integers that satisfy the system of equations
a
3
+
a
2
+
b
2
=
0
a^3 + a^2 + b^2 = 0
a
3
+
a
2
+
b
2
=
0
b
3
+
b
2
+
c
2
=
0
b^3 + b^2 + c^2 = 0
b
3
+
b
2
+
c
2
=
0
c
3
+
c
2
+
a
2
=
0
c^3 + c^2 + a^2 = 0
c
3
+
c
2
+
a
2
=
0
p4. In a remote country the Oenemense language is used, which is subject to the following rules: (i) The alphabet consists only of the letters O, N, M. (ii) Every syllable consists of the three letters O, N, M in some order. (iii) Every word consists of one or more syllables, so that there are no consecutive vowels. For example, ONMNOM is a valid word, but NMOONM is not. Show that for all
n
≥
2
n \ge 2
n
≥
2
the number of words of
n
n
n
syllables is a multiple of
4
4
4
. p5. A die is a cube with side
1
1
1
with the numbers from
1
1
1
to
6
6
6
on each of its faces, so that each number appears only once, and the sum of the numbers on opposite faces is
7
7
7
. A cube of
3
×
3
3 \times 3
3
×
3
using 27 dice, and then the sum of all the numbers that appear above it's faces is calculated. Determine the possible values of this sum.