2005 USAMO Problems
Contents
Day 1
Problem 1
Determine all composite positive integers for which it is possible to arrange all divisors of
that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
Problem 2
Prove that the
system
has no solutions in integers
,
, and
.
Problem 3
Let be an acute-angled triangle, and let
and
be two points on side
. Construct point
in such a way that convex quadrilateral
is cyclic,
, and
and
lie on opposite sides of line
. Construct point
in such a way that convex quadrilateral
is cyclic,
, and
and
lie on opposite sides of line
. Prove that points
, and
lie on a circle.