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 19:50, 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.