Difference between revisions of "2002 AIME II Problems/Problem 4"
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== Problem == | == Problem == | ||
| + | Patio blocks that are hexagons <math>1</math> unit on a side are used to outline a garden by placing the blocks edge to edge with <math>n</math> on each side. The diagram indicates the path of blocks around the garden when <math>n=5</math>. | ||
| − | == Solution == | + | [[Image:AIME 2002 II Problem 4.gif]] |
| + | |||
| + | If <math>n=202</math>, then the area of the garden enclosed by the path, not including the path itself, is <math>m\left(\sqrt3/2\right)</math> square units, where <math>m</math> is a positive integer. Find the remainder when <math>m</math> is divided by <math>1000</math>. | ||
| + | |||
| + | == Solution 1== | ||
| + | When <math>n>1</math>, the path of blocks has <math>6(n-1)</math> blocks total in it. When <math>n=1</math>, there is just one lonely block. Thus, the area of the garden enclosed by the path when <math>n=202</math> is | ||
| + | |||
| + | <cmath>(1+6+12+18+\cdots +1200)A=(1+6(1+2+3...+200))A</cmath>, | ||
| + | |||
| + | where <math>A</math> is the area of one block. Then, because <math>n(n+1)/2</math> is equal to the sum of the first <math>n</math> integers: | ||
| + | |||
| + | <cmath>(1+6(1+2+3...+200))=(1+6((200)(201)/2))A=120601A</cmath>. | ||
| + | |||
| + | Since <math>A=\dfrac{3\sqrt{3}}{2}</math>, the area of the garden is | ||
| + | |||
| + | <cmath>120601\cdot \dfrac{3\sqrt{3}}{2}=\dfrac{361803\sqrt{3}}{2}</cmath>. | ||
| + | |||
| + | <math>m=361803</math>, <math>\dfrac{m}{1000}=361</math> Remainder <math>\boxed{803}</math>. | ||
| + | |||
| + | == Solution 2== | ||
| + | Note that this is just the definition for a centered hexagonal number, and the formula for <math>(n-1)^{th}</math> term is <math>3n(n+1)+1</math>. Applying this for <math>200</math> as we want the inner area gives <math>120601</math>. Then continue as above. | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=2002|n=II|num-b=3|num-a=5}} | |
| + | |||
| + | [[Category: Intermediate Geometry Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 06:28, 13 September 2020
Contents
Problem
Patio blocks that are hexagons 
unit on a side are used to outline a garden by placing the blocks edge to edge with 
on each side. The diagram indicates the path of blocks around the garden when 
.
If 
, then the area of the garden enclosed by the path, not including the path itself, is 
square units, where 
is a positive integer. Find the remainder when is divided by 
.
Solution 1
When 
, the path of blocks has 
blocks total in it. When 
, there is just one lonely block. Thus, the area of the garden enclosed by the path when is
![\[(1+6+12+18+\cdots +1200)A=(1+6(1+2+3...+200))A\]](http://latex.artofproblemsolving.com/e/c/a/ecad98b857944eb31fe872e4829687a8ce265342.png)
,
where 
is the area of one block. Then, because 
is equal to the sum of the first integers:
![\[(1+6(1+2+3...+200))=(1+6((200)(201)/2))A=120601A\]](http://latex.artofproblemsolving.com/2/3/1/2319ef0a32ca4887a6f58234c250d5689530db33.png)
.
Since 
, the area of the garden is
![\[120601\cdot \dfrac{3\sqrt{3}}{2}=\dfrac{361803\sqrt{3}}{2}\]](http://latex.artofproblemsolving.com/1/2/8/1283821e2343826447216d84e3c55534912c388c.png)
.
, 
Remainder 
.
Solution 2
Note that this is just the definition for a centered hexagonal number, and the formula for 
term is 
. Applying this for 
as we want the inner area gives 
. Then continue as above.
See also
| 2002 AIME II (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. 
