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National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2012 El Salvador Correspondence
2012 El Salvador Correspondence
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El Salvador Correspondence
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2012 El Salvador Correspondence / Qualifying NMO XII
p1. Determine all three-digit numbers
X
Y
Z
‾
\overline{XYZ}
X
Y
Z
that meet the equality
X
Y
Z
‾
+
Y
X
Z
‾
=
W
W
Z
Z
‾
\overline{XYZ} + \overline{YXZ} = \overline{WWZZ}
X
Y
Z
+
Y
XZ
=
WW
ZZ
where each letter represents a digit, equal letters represent equal digits, and different letters represent different digits. p2. A machine for numbering pages is ruined, the problem it has is that it does not print the numbers that are multiples of
3
3
3
nor those that are multiples of
5
5
5
. The numbering on the pages of a book is the following:
1
,
2
,
4
,
7
,
8
,
11
,
.
.
.
1, 2, 4, 7, 8, 11, ...
1
,
2
,
4
,
7
,
8
,
11
,
...
If the last issue printed in this book is
2012
2012
2012
, determine the number of pages in the book. p3. Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be three positive numbers that satisfy
x
+
1
y
=
7
3
x+\frac{1}{y}=\frac73
x
+
y
1
=
3
7
y
+
1
z
=
1
y+\frac{1}{z}=1
y
+
z
1
=
1
z
+
1
x
=
4
z+\frac{1}{x}=4
z
+
x
1
=
4
Determine the product
x
y
z
xyz
x
yz
. p4. Let
A
B
C
ABC
A
BC
be a triangle right at
C
C
C
, draw the altitude
C
D
CD
C
D
. It is known that the area of the triangle
A
D
C
ADC
A
D
C
is one fourth part of the area of the triangle
A
B
C
ABC
A
BC
. Determine the measures of the angles of triangle
A
B
C
ABC
A
BC
p5. Determine for what value
n
n
n
it is possible to form with
9
9
9
people,
n
n
n
groups of
3
3
3
people each so that each pair of people is in exactly one group. Show one of the possible group configurations.