Difference between revisions of "1982 USAMO Problems"
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Revision as of 00:44, 28 August 2011
Contents
[hide]Problem 1
A graph has points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to
points?
Problem 2
Show that if are positive integers such that
for all real
with sum
, then
or
.
Problem 3
is a point inside the equilateral triangle
.
is a point inside
. Show that
Problem 4
Show that there is a positive integer such that, for every positive integer
,
is composite.
Problem 5
is the center of a sphere
. Points
are inside
,
is perpendicular to
and
, and there are two spheres through
, and
which touch
. Show that the sum of their radii equals the radius of
.