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El Salvador Correspondence
2014 El Salvador Correspondence
2014 El Salvador Correspondence
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El Salvador Correspondence
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2014 El Salvador Correspondence / Qualifying NMO XIV
p1. There are
11
11
11
large boxes. Some of them contain, each one,
8
8
8
medium boxes and the rest are empty. Also, some of the medium boxes contain, each one,
8
8
8
small boxes and the rest are empty. All the small boxes are empty. If there are
102
102
102
empty boxes, determine the total number of boxes. p2. If the lengths
a
,
b
a, b
a
,
b
and
c
c
c
of the sides of a triangle satisfy the conditions
a
+
b
−
c
=
2
a + b - c = 2
a
+
b
−
c
=
2
2
a
b
−
c
2
=
4
2ab - c^2 = 4
2
ab
−
c
2
=
4
prove that the triangle is equilateral. p3. A square of area
A
A
A
is divided into
102
102
102
smaller squares,
101
101
101
of which have sides of length
1
1
1
. Determine all possible values for
A
A
A
. p4. In the figure, the ratio of the radius of the circular sector to the radius of the inner circle is
3
:
1
3: 1
3
:
1
. If the circle is tangent to the edges of the sector, determine the ratio of its areas. https://cdn.artofproblemsolving.com/attachments/2/c/6636dee0e9a10da8bde10f06ce297e9faca074.png p5. A convex polyhedron has
12
12
12
squares,
8
8
8
hexagons, and
6
6
6
octagons for faces. At every vertex exactly one square, one hexagon and one octagon concur. Consider the line segments that join two vertices of the polyhedron and determine how many of them are neither edges nor contained in some face of the polyhedron.