Difference between revisions of "1991 USAMO Problems"
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* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1991.tex 1991 USAMO Problems (TEX)] | * [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1991.tex 1991 USAMO Problems (TEX)] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=1991 1991 USAMO Problems on the resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=1991 1991 USAMO Problems on the resources page] | ||
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Revision as of 20:50, 3 July 2013
Problems from the 1991 USAMO. There were five questions administered in one three-and-a-half-hour session.
Problem 1
In triangle angle
is twice angle
angle
is obtuse, and the three side lengths
are integers. Determine, with proof, the minimum possible perimeter.
Problem 2
For any nonempty set of numbers, let
and
denote the sum and product, respectively, of the elements of
. Prove that
where "
" denotes a sum involving all nonempty subsets
of
.
Problem 3
Show that, for any fixed integer the sequence
is eventually constant.
[The tower of exponents is defined by . Also
means the remainder which results from dividing
by
.]
Problem 4
Let where
and
are positive integers. Prove that
.
[You may wish to analyze the ratio for real
and integer
.]
Problem 5
Let be an arbitrary point on side
of a given triangle
and let
be the interior point where
intersects the external common tangent to the incircles of triangles
and
. As
assumes all positions between
and
, prove that the point
traces the arc of a circle.
Resources
1991 USAMO (Problems • Resources) | ||
Preceded by 1990 USAMO |
Followed by 1992 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.