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Title: A Putnam Exam Challenge Post by: zhangx on Nov 30, 2011 A dart thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Write your answer in the form (a*sqrt(b)+c)/d, where a,b,c and d are positive integers.
Title: Re: A Putnam Exam Challenge Post by: BearPro on Nov 30, 2011 (http://dl.dropbox.com/u/35279697/Screenshot-2.jpg)
We need the perpendicular distance to the side. The distances are equal when PO = PQ The locus of P is a parabola, contained between diagonals OA and OB. So, If you look at shape of the area that is closer to the center than to any edge. It will be like this (http://dl.dropbox.com/u/35279697/Screenshot-3.jpg) Solving it (http://dl.dropbox.com/u/35279697/Screenshot-4.jpg) (http://dl.dropbox.com/u/35279697/Screenshot-5.jpg) |