Party optimization problem
by shiningsunnyday, Oct 29, 2017, 5:00 PM
You arrive in a room with 
number of girls, with attractiveness 
respectively and at locations 
Where should you stand to "optimize" your interaction, governed by the heuristic that the distance at which you're likely to stand from a girl is solely dependent on and inversely proportional to her attractiveness?
number of girls, with attractiveness 
respectively and at locations 
Where should you stand to "optimize" your interaction, governed by the heuristic that the distance at which you're likely to stand from a girl is solely dependent on and inversely proportional to her attractiveness?
![\[\frac{r_1}{r_2} = \frac{A_2}{A_1}\]](http://latex.artofproblemsolving.com/0/6/9/06908fbfed020616b99272f08ff7374b977f4e26.png)
where 
is your distance from girls one and two. By Geo 3 the locus of points is actually the 
we can replace 
respectively, so we have ![\[\frac{x-x_1}{x_2-x} = \frac{A_2}{A_1}\]](http://latex.artofproblemsolving.com/1/a/6/1a670f476a305d460bda923b1ee087e6b1a569a4.png)
which arranges to 
![\[\left(\frac{x_1A_1+x_2A_2+x_3A_3}{A_1+A_2+A_3}, \frac{y_1A_1+y_2A_2+y_3A_3}{A_1+A_2+A_3} \right)\]](http://latex.artofproblemsolving.com/1/9/4/19428143bf8af4a10fbe708cf3ee6e5a70e8d910.png)
but consider the case when 
We shouldn't expect to closer to one girl than the other, so we stand at a point that's equidistant to each girl, so it's expected we should stand on the circumcenter.
The idea is that the locus of points we're allowed to stand on given 
is a perpendicular line going through the center of mass of girls one and two. But clearly these three perpendicular lines don't have to intersect.
if 
this will only work if the polygon is cyclic.This post has been edited 1 time. Last edited by shiningsunnyday, Oct 30, 2017, 8:19 AM
May 2018
April 2018