Difference between revisions of "2000 AIME I Problems/Problem 4"
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== Problem == | == Problem == | ||
− | The diagram shows a rectangle that has been dissected into nine non-overlapping | + | The diagram shows a [[rectangle]] that has been dissected into nine non-overlapping [[square]]s. Given that the width and the height of the rectangle are relatively prime positive integers, find the [[perimeter]] of the rectangle. |
− | + | <asy>draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); | |
draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); | draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); | ||
draw((34,36)--(34,45)--(25,45)); | draw((34,36)--(34,45)--(25,45)); | ||
draw((36,36)--(36,38)--(34,38)); | draw((36,36)--(36,38)--(34,38)); | ||
draw((36,38)--(41,38)); | draw((36,38)--(41,38)); | ||
− | draw((34,45)--(41,45));</asy | + | draw((34,45)--(41,45));</asy> |
− | == 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. | |
− | The picture shows that < | + | The picture shows that |
+ | <cmath>\begin{align*} | ||
+ | a_1+a_2 &= a_3\\ | ||
+ | a_1 + a_3 &= a_4\\ | ||
+ | a_3 + a_4 &= a_5\\ | ||
+ | a_4 + a_5 &= a_6\\ | ||
+ | a_2 + a_3 + a_5 &= a_7\\ | ||
+ | a_2 + a_7 &= a_8\\ | ||
+ | a_1 + a_4 + a_6 &= a_9\\ | ||
+ | a_6 + a_9 &= a_7 + a_8.\end{align*}</cmath> | ||
− | + | Expressing all terms 3 to 9 in terms of <math>a_1</math> and <math>a_2</math> and substituting their expanded forms into the previous equation will give the expression <math>5a_1 = 2a_2</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 1.2 (more detail) == | ||
+ | We can just list the equations: | ||
+ | <cmath>\begin{align*} | ||
+ | s_3 &= s_1 + s_2 \\ | ||
+ | s_4 &= s_3 + s_1 \\ | ||
+ | s_5 &= s_4 + s_3 \\ | ||
+ | s_6 &= s_5 + s_4 \\ | ||
+ | s_7 &= s_5 + s_3 + s_2 \\ | ||
+ | s_8 &= s_7 + s_2 \\ | ||
+ | s_9 &= s_8 + s_2 - s_1 \\ | ||
+ | s_9 + s_8 &= s_7 + s_6 + s_5 \end{align*}</cmath>We can then write each <math>s_i</math> in terms of <math>s_1</math> and <math>s_2</math> as follows | ||
+ | <cmath>\begin{align*} | ||
+ | s_4 &= 2s_1 + s_2 \\ | ||
+ | s_5 &= 3s_1 +2s_2 \\ | ||
+ | s_6 &= 5s_1 + 3s_2 \\ | ||
+ | s_7 &= 4s_1 + 4s_2 \\ | ||
+ | s_8 &= 4s_1 + 5s_2 \\ | ||
+ | s_9 &= 3s_1 + 6s_2 \\ | ||
+ | \end{align*}</cmath> | ||
+ | Since <math>s_9 + s_8 = s_7 + s_6 + s_5 \implies (3s_1 + 6s_2) + (4s_1 + 5s_2) = (4s_1 + 4s_2) + (5s_1 + 3s_2) + (3s_1 + 2s_2),</math> <cmath>2s_2 = 5s_1 \implies \frac{2}{5}s_2 = s_1.</cmath>Since the side lengths of the rectangle are relatively prime, we can see that <math>s_1 = 2</math> and <math>s_2 = 5.</math> Therefore, <math>2(2s_9 + s_6 + s_8) = 30s_1 + 40s_2 = \boxed{260}.</math> | ||
+ | ~peelybonehead | ||
+ | |||
+ | ==Solution 2 Length-chasing (Angle-chasing but for side lengths) == | ||
+ | |||
+ | We set the side length of the smallest square to 1, and set the side length of square <math>a_4</math> in the previous question to a. We do some "side length chasing" and get <math>4a - 4 = 2a + 5</math>. Solving, we get <math>a = 4.5</math> and the side lengths are <math>61</math> and <math>69</math>. Thus, the perimeter of the rectangle is <math>2(61 + 69) = \boxed{260}.</math> | ||
== See also == | == See also == | ||
{{AIME box|year=2000|n=I|num-b=3|num-a=5}} | {{AIME box|year=2000|n=I|num-b=3|num-a=5}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 00:31, 19 January 2024
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 1.2 (more detail)
We can just list the equations:
We can then write each
in terms of
and
as follows
Since
Since the side lengths of the rectangle are relatively prime, we can see that
and
Therefore,
~peelybonehead
Solution 2 Length-chasing (Angle-chasing but for side lengths)
We set the side length of the smallest square to 1, and set the side length of square in the previous question to a. We do some "side length chasing" and get
. Solving, we get
and the side lengths are
and
. Thus, the perimeter of the rectangle 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.