Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 6"
Fireclaw105 (talk | contribs) (Created page with "'''Solution''' We claim that all angles in triangle ABC have to be less than 120 in order for P to be inside the equilateral triangle.") |
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| − | + | ==Problem== | |
| + | Three points <math>A,B,C</math> are chosen at random on a circle. The probability that there exists a point <math>P</math> inside an equilateral triangle <math>A_1B_1C_1</math> such that <math>PA_1=BC,PB_1=AC,PC_1=AB</math> can be expressed in the form <math>\frac{m} {n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
| − | + | ==Solution== | |
| + | The problem asks for the probability that point <math>P</math> is inside an equilateral triangle <math>A_1B_1C_1</math>. Let <math>x</math>, <math>y</math>, and <math>z</math> be the three distances from point <math>P</math> to each of the vertices, with <math>x</math> being the longest distance. Let's consider the case in which point <math>P</math> is actually on the line: | ||
| + | <asy> | ||
| + | unitsize(1cm); | ||
| + | draw((0, 18*sqrt(3))--(18, 0)); | ||
| + | draw((18, 0)--(36, 18*sqrt(3))); | ||
| + | <\asy> | ||
Revision as of 18:01, 7 July 2019
Problem
Three points 
are chosen at random on a circle. The probability that there exists a point 
inside an equilateral triangle 
such that 
can be expressed in the form 
where 
are relatively prime positive integers. Find 
Solution
The problem asks for the probability that point is inside an equilateral triangle . Let 
, 
, and 
be the three distances from point to each of the vertices, with being the longest distance. Let's consider the case in which point is actually on the line:
<asy>
unitsize(1cm);
draw((0, 18*sqrt(3))--(18, 0));
draw((18, 0)--(36, 18*sqrt(3)));
<\asy>
