1991 USAMO Problems
In triangle angle is twice angle angle is obtuse, and the three side lengths are integers. Determine, with proof, the minimum possible perimeter.
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 .
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 .]
Let where and are positive integers. Prove that .
[You may wish to analyze the ratio for real and integer .]
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.
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