Difference between revisions of "2000 AIME I Problems/Problem 4"
Danielguo94 (talk | contribs) (→Solution) |
(→Solution) |
||
Line 12: | Line 12: | ||
Call the squares' side lengths from smallest to largest <math>a_1,\ldots,a_9</math>, and let <math>l,w</math> represent the dimensions of the rectangle. | Call the squares' side lengths from smallest to largest <math>a_1,\ldots,a_9</math>, and let <math>l,w</math> represent the dimensions of the rectangle. | ||
− | The picture shows that | + | The picture shows that |
− | <cmath>\begin{ | + | <cmath>\begin{align*} |
− | a_1+a_2 &= | + | a_1+a_2 &= a_3\ |
− | a_1 + a_3 &= | + | a_1 + a_3 &= a_4\ |
− | a_3 + a_4 &= | + | a_3 + a_4 &= a_5\ |
− | a_4 + a_5 &= | + | a_4 + a_5 &= a_6\ |
− | a_2 + a_3 + a_5 &= | + | a_2 + a_3 + a_5 &= a_7\ |
− | a_2 + a_7 &= | + | a_2 + a_7 &= a_8\ |
− | a_1 + a_4 + a_6 &= | + | a_1 + a_4 + a_6 &= a_9\ \intertext{and} |
− | a_6 + a_9 &= | + | a_6 + a_9 &= a_7 + a_8\end{align*}</cmath> |
− | With a bit of trial and error and some arithmetic, we can use | + | With a bit of trial and error and some arithmetic, we can use these equations to find that <math>5a_1 = 2a_2</math>; we can guess that <math>a_1 = 2</math>. Then solving gives <math>a_9 = 36</math>, <math>a_6=25</math>, <math>a_8 = 33</math>, which gives us <math>l=61,w=69</math>. These numbers are relatively prime, as desired. (If we started with <math>a_1</math> odd, the resulting sides would not be integers and we would need to scale up by a factor of <math>2</math> to make them integers; if we started with <math>a_1 > 2</math> even, the resulting dimensions would not be relatively prime and we would need to scale down.) The perimeter is <math>2(61)+2(69)=\boxed{260}</math>. |
== See also == | == See also == |
Revision as of 07:55, 26 January 2012
Problem
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
Solution
Call the squares' side lengths from smallest to largest , and let represent the dimensions of the rectangle.
The picture shows that
With a bit of trial and error and some arithmetic, we can use these equations to find that ; we can guess that . Then solving gives , , , which gives us . These numbers are relatively prime, as desired. (If we started with odd, the resulting sides would not be integers and we would need to scale up by a factor of to make them integers; if we started with even, the resulting dimensions would not be relatively prime and we would need to scale down.) The perimeter is .
See also
2000 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |