1995 AIME Problems/Problem 1
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
Solution
The sum of the areas of the squares if they were not interconnected is a geometric sequence:
Then subtract the areas of the intersections, which is :
The majority of the terms cancel, leaving , which simplifies down to . Thus, .
Alternatively, take the area of the first square and add of the areas of the remaining squares. This results in , which when simplified will produce the same answer.
See also
1995 AIME (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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