Difference between revisions of "2000 AIME I Problems/Problem 4"
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− | == Solution == | + | == Solution 1 == |
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. | ||
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We can guess that <math>a_1 = 2</math>. (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.) 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. The perimeter is <math>2(61)+2(69)=\boxed{260}</math>. | We can guess that <math>a_1 = 2</math>. (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.) 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. The perimeter is <math>2(61)+2(69)=\boxed{260}</math>. | ||
+ | |||
+ | ==Solution 2 Length-chasing (Angle-chasing but for side lengths) == | ||
== See also == | == See also == |
Revision as of 13:17, 11 January 2023
Contents
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 1
Call the squares' side lengths from smallest to largest , and let
represent the dimensions of the rectangle.
The picture shows that
Expressing all terms 3 to 9 in terms of and
and substituting their expanded forms into the previous equation will give the expression
.
We can guess that . (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.) Then solving gives
,
,
, which gives us
. These numbers are relatively prime, as desired. The perimeter is
.
Solution 2 Length-chasing (Angle-chasing but for side lengths)
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.