Difference between revisions of "1995 AIME Problems/Problem 1"
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Square <math>\displaystyle S_{1}</math> is <math>1\times 1.</math> For <math>i\ge 1,</math> the lengths of the sides of square <math>\displaystyle S_{i+1}</math> are half the lengths of the sides of square <math>\displaystyle S_{i},</math> two adjacent sides of square <math>\displaystyle S_{i}</math> are perpendicular bisectors of two adjacent sides of square <math>\displaystyle S_{i+1},</math> and the other two sides of square <math>\displaystyle S_{i+1},</math> are the perpendicular bisectors of two adjacent sides of square <math>\displaystyle S_{i+2}.</math> The total area enclosed by at least one of <math>\displaystyle S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</math> can be written in the form <math>\displaystyle m/n,</math> where <math>\displaystyle m</math> and <math>\displaystyle n</math> are relatively prime positive integers. Find <math>\displaystyle m-n.</math> | Square <math>\displaystyle S_{1}</math> is <math>1\times 1.</math> For <math>i\ge 1,</math> the lengths of the sides of square <math>\displaystyle S_{i+1}</math> are half the lengths of the sides of square <math>\displaystyle S_{i},</math> two adjacent sides of square <math>\displaystyle S_{i}</math> are perpendicular bisectors of two adjacent sides of square <math>\displaystyle S_{i+1},</math> and the other two sides of square <math>\displaystyle S_{i+1},</math> are the perpendicular bisectors of two adjacent sides of square <math>\displaystyle S_{i+2}.</math> The total area enclosed by at least one of <math>\displaystyle S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</math> can be written in the form <math>\displaystyle m/n,</math> where <math>\displaystyle m</math> and <math>\displaystyle n</math> are relatively prime positive integers. Find <math>\displaystyle m-n.</math> | ||
| + | [[Image:AIME 1995 Problem 1.png]] | ||
== Solution == | == Solution == | ||
== See also == | == See also == | ||
* [[1995 AIME Problems]] | * [[1995 AIME Problems]] | ||
Revision as of 20:49, 21 January 2007
Problem
Square 
is 
For 
the lengths of the sides of square 
are half the lengths of the sides of square 
two adjacent sides of square 
are perpendicular bisectors of two adjacent sides of square 
and the other two sides of square are the perpendicular bisectors of two adjacent sides of square 
The total area enclosed by at least one of 
can be written in the form 
where 
and 
are relatively prime positive integers. Find 
