Difference between revisions of "1994 USAMO Problems"
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==Problem 2== | ==Problem 2== | ||
− | + | 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 | |
− | 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? |
− | are red, blue, red, blue, red, blue, red, yellow, blue? | ||
[[1994 USAMO Problems/Problem 2|Solution]] | [[1994 USAMO Problems/Problem 2|Solution]] |
Revision as of 11:53, 12 April 2011
Problem 1
Let be positive integers, no two consecutive, and let for . Prove that, for each positive integer the interval contains at least one perfect square.
Problem 2
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?
Problem 3
A convex hexagon is inscribed in a circle such that and diagonals , , and are concurrent. Let be the intersection of and . Prove that .
Problem 4
Let be a sequence of positive real numbers satisfying for all . Prove that, for all
Problem 5
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 .
Resources
1994 USAMO (Problems • Resources) | ||
Preceded by 1993 USAMO |
Followed by 1995 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |