Difference between revisions of "1994 USAMO Problems"
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[[1994 USAMO Problems/Problem 1|Solution]] | [[1994 USAMO Problems/Problem 1|Solution]] | ||
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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]] | ||
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==Problem 5== | ==Problem 5== | ||
− | + | Let <math>\, |U|, \, \sigma(U) \,</math> and <math>\, \pi(U) \,</math> denote the number of elements, the sum, and the product, respectively, of a finite set <math>\, U \,</math> of positive integers. (If <math>\, U \,</math> is the empty set, <math>\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1</math>.) Let <math>\, S \,</math> be a finite set of positive integers. As usual, let <math>\, \binom{n}{k} \,</math> denote <math>\, n! \over k! \, (n-k)!</math>. Prove that <cmath> \sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S) </cmath> for all integers <math>\, m \geq \sigma(S)</math>. | |
− | Let <math>\, |U|, \, \sigma(U) \,</math> and <math>\, \pi(U) \,</math> denote the number of elements, the sum, and the product, respectively, of a finite set <math>\, U \,</math> of positive integers. (If <math>\, U \,</math> is the empty set, <math>\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1</math>.) Let <math>\, S \,</math> be a finite set of positive integers. As usual, let <math>\, \binom{n}{k} \,</math> denote <math>\, n! \over k! \, (n - k)!</math>. Prove that | ||
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− | <cmath>\sum_{U \subseteq S} ( - 1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S)</cmath> | ||
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− | for all integers <math>\, m \geq \sigma(S)</math>. | ||
[[1994 USAMO Problems/Problem 5|Solution]] | [[1994 USAMO Problems/Problem 5|Solution]] | ||
− | == | + | == See Also == |
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{{USAMO box|year=1994|before=[[1993 USAMO]]|after=[[1995 USAMO]]}} | {{USAMO box|year=1994|before=[[1993 USAMO]]|after=[[1995 USAMO]]}} | ||
− | + | {{MAA Notice}} | |
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Latest revision as of 07:58, 19 July 2016
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 .
See Also
1994 USAMO (Problems • Resources) | ||
Preceded by 1993 USAMO |
Followed by 1995 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.