Difference between revisions of "2002 AIME II Problems/Problem 4"
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Revision as of 19:36, 4 July 2013
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
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
![\[(1+6+12+18+\cdots +1200)A\]](http://latex.artofproblemsolving.com/a/d/2/ad2ac6c45d596e5199074194258210022ef4b942.png)
,
where 
is the area of one block. 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 
.
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. 
