1994 USAMO Problems
Let be positive integers, no two consecutive, and let for . Prove that, for each positive integer the interval contains at least one perfect square.
The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, ... red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue, ... red, yellow, blue?
A convex hexagon is inscribed in a circle such that and diagonals , , and are concurrent. Let be the intersection of and . Prove that .
Let be a sequence of positive real numbers satisfying for all . Prove that, for all
Let and denote the number of elements, the sum, and the product, respectively, of a finite set of positive integers. (If is the empty set, .) Let be a finite set of positive integers. As usual, let denote . Prove that for all integers .
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