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National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2003 El Salvador Correspondence
2003 El Salvador Correspondence
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El Salvador Correspondence
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2003 El Salvador Correspondence / Qualifying NMO III
p1. Determine the smallest natural number that has the property that it's cube ends in
888
888
888
. p2. Triangle
A
B
C
ABC
A
BC
is isosceles with
A
C
=
B
C
AC = BC
A
C
=
BC
. The angle bisector at
A
A
A
intercepts side
B
C
BC
BC
at the point
D
D
D
and the bisector of the angle at
C
C
C
intercepts side
A
B
AB
A
B
at
E
E
E
. If
A
D
=
2
C
E
AD = 2CE
A
D
=
2
CE
, find the measure of the angles of triangle
A
B
C
ABC
A
BC
. p3.In the accompanying figure, the circle
C
C
C
is tangent to the quadrant
A
O
B
AOB
A
OB
at point
S
S
S
. AP is tangent to
C
C
C
at
P
P
P
and
O
Q
OQ
OQ
is tangent at
Q
Q
Q
. Calculate the length
A
P
AP
A
P
as a function of the radius
R
R
R
of the quadrant
A
O
B
AOB
A
OB
. https://cdn.artofproblemsolving.com/attachments/3/c/d34774c3c6c33d351316574ca3f7ade54e6441.png p4. Let
A
A
A
be a set with seven or more natural numbers. Show that there must be two numbers in
A
A
A
with the property that either its sum or its difference is divisible by ten. Show that the property can fail if set
A
A
A
has fewer than seven elements. p5. An
n
n
n
-sided convex polygon is such that there is no common point for any three of its diagonals. Determine the number of triangles that are formed such that two of their vertices are vertices of the polygon and the third is an intersection of two diagonals.