Difference between revisions of "1995 AIME Problems"

Line 1: Line 1:
 
== Problem 1 ==
 
== Problem 1 ==
 
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]]
  
 
[[1995 AIME Problems/Problem 1|Solution]]
 
[[1995 AIME Problems/Problem 1|Solution]]

Revision as of 21:49, 21 January 2007

Problem 1

Square $\displaystyle S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $\displaystyle S_{i+1}$ are half the lengths of the sides of square $\displaystyle S_{i},$ two adjacent sides of square $\displaystyle S_{i}$ are perpendicular bisectors of two adjacent sides of square $\displaystyle S_{i+1},$ and the other two sides of square $\displaystyle S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $\displaystyle S_{i+2}.$ The total area enclosed by at least one of $\displaystyle S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $\displaystyle m/n,$ where $\displaystyle m$ and $\displaystyle n$ are relatively prime positive integers. Find $\displaystyle m-n.$

AIME 1995 Problem 1.png

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

Invalid username
Login to AoPS