Difference between revisions of "1994 AIME Problems/Problem 15"

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== Problem ==
 
== Problem ==
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Given a point <math>P^{}_{}</math> on a triangular piece of paper <math>ABC,\,</math> consider the creases that are formed in the paper when <math>A, B,\,</math> and <math>C\,</math> are folded onto <math>P.\,</math>  Let us call <math>P_{}^{}</math> a fold point of <math>\triangle ABC\,</math> if these creases, which number three unless <math>P^{}_{}</math> is one of the vertices, do not intersect.  Suppose that <math>AB=36, AC=72,\,</math> and <math>\angle B=90^\circ.\,</math>  Then the area of the set of all fold points of <math>\triangle ABC\,</math> can be written in the form <math>q\pi-r\sqrt{s},\,</math> where <math>q, r,\,</math> and <math>s\,</math> are positive integers and <math>s\,</math> is not divisible by the square of any prime.  What is <math>q+r+s\,</math>?
  
 
== Solution ==
 
== Solution ==
 
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{{solution}}
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== See also ==
 
== See also ==
* [[1994 AIME Problems/Problem 14 | Previous problem]]
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{{AIME box|year=1994|num-b=14|after=Last question}}
* [[1994 AIME Problems]]
 

Revision as of 22:47, 28 March 2007

Problem

Given a point $P^{}_{}$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P_{}^{}$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P^{}_{}$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,\,$ and $\angle B=90^\circ.\,$ Then the area of the set of all fold points of $\triangle ABC\,$ can be written in the form $q\pi-r\sqrt{s},\,$ where $q, r,\,$ and $s\,$ are positive integers and $s\,$ is not divisible by the square of any prime. What is $q+r+s\,$?

Solution

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See also

1994 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
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