2002 AIME II Problems/Problem 4

Problem

Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$ .

If $n=202$ , then the area of the garden enclosed by the path, not including the path itself, is $m\left(\sqrt3/2\right)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000$ .

Solution

When $n>1$ , the path of blocks has $6(n-1)$ blocks total in it. When $n=1$ , there is just one lonely block. Thus, the area of the garden enclosed by the path when $n=202$ is

$(1+6+12+18+\cdots +1200)A=(1+6(1+2+3...+200))A$

Then, because n(n+1)/2 is equal to the sum of the first n integers,

$(1+6(1+2+3...+200))=(1+6((200)(201)/2))A=120601A$

where $A$ is the area of one block. Since $A=\dfrac{3\sqrt{3}}{2}$ , the area of the garden is

$120601\cdot \dfrac{3\sqrt{3}}{2}=\dfrac{361803\sqrt{3}}{2}$ .

$m=361803$ , $\dfrac{m}{1000}=361$ Remainder $\boxed{803}$ .

2002 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2002 AIME II Problems/Problem 4

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