2002 AIME II Problems/Problem 4
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
,
where is the area of one block. Then, because is equal to the sum of the first integers:
.
Since , the area of the garden is
.
, Remainder .
Solution 2
Note that this is just the definition for a centered hexagonal number, and directly applying 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 |
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