Mock AIME 2 2006-2007/Problem 7
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Problem
A right circular cone of base radius 
cm and slant height 
cm is given. 
is a point on the circumference of the base and the shortest path from around the cone and back is drawn (see diagram). If the length of this path is 
where 
and 
are relatively prime positive integers, find 
Solution
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We begin by noticing that that the shortest path from a point on the base to and around the cone is the perpendicular from the aforementioned point to the slant line. We can now construct two equations based on this fact. Let 
be the length from the point on the other end of the diameter from to the point at which is perpendicular to the slant. Let 
now be the diameter of the shaded circle. We now have two equations : 
and 
. We can tell from this equation that 
, so 
. From this we can deduce that y = 
.
