Difference between revisions of "2005 AIME I Problems/Problem 14"
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Revision as of 08:05, 31 July 2007
Problem
Consider the points 
and 
There is a unique square 
such that each of the four points is on a different side of 
Let 
be the area of Find the remainder when 
is divided by 1000.
Solution
Let 
denote a normal vector of the side containing 
. The lines containing the sides of the square have the form 
, 
, 
and 
. The lines form a square, so the distance between 
and the line through equals the distance between 
and the line through 
, hence 
, or 
. We can take 
and 
. So the side of the square is 
, the area is 
, and the answer to the problem is 
.
