Difference between revisions of "2003 AIME I Problems"
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== Problem 13 == | == Problem 13 == | ||
+ | Let <math> N </math> be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when <math> N </math> is divided by 1000. | ||
[[2003 AIME I Problems/Problem 13|Solution]] | [[2003 AIME I Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | The decimal representation of <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers and <math> m < n, </math> contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of <math> n </math> for which this is possible. | ||
[[2003 AIME I Problems/Problem 14|Solution]] | [[2003 AIME I Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | In <math> \triangle ABC, AB = 360, BC = 507, </math> and <math> CA = 780. </math> Let <math> M </math> be the midpoint of <math> \overline{CA}, </math> and let <math> D </math> be the point on <math> \overline{CA} </math> such that <math> \overline{BD} </math> bisects angle <math> ABC. </math> Let <math> F </math> be the point on <math> \overline{BC} </math> such that <math> \overline{DF} \perp \overline{BD}. </math> Suppose that <math> \overline{DF} </math> meets <math> \overline{BM} </math> at <math> E. </math> The ratio <math> DE: EF </math> can be written in the form <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. Find <math> m + n. </math> | ||
[[2003 AIME I Problems/Problem 15|Solution]] | [[2003 AIME I Problems/Problem 15|Solution]] |
Revision as of 18:13, 9 July 2006
Contents
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Let be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when is divided by 1000.
Problem 14
The decimal representation of where and are relatively prime positive integers and contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of for which this is possible.
Problem 15
In and Let be the midpoint of and let be the point on such that bisects angle Let be the point on such that Suppose that meets at The ratio can be written in the form where and are relatively prime positive integers. Find