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Difference between revisions of "2003 AIME I Problems"

 
Line 48: Line 48:
  
 
== 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 19:13, 9 July 2006

Problem 1

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Let $N$ 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 $N$ is divided by 1000.

Solution

Problem 14

The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.

Solution

Problem 15

In $\triangle ABC, AB = 360, BC = 507,$ and $CA = 780.$ Let $M$ be the midpoint of $\overline{CA},$ and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC.$ Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}.$ Suppose that $\overline{DF}$ meets $\overline{BM}$ at $E.$ The ratio $DE: EF$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

See also

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