Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.9 AMC 10/12 Find the answer

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 $n=5$ .
AIME 2002 II Problem 4.gif
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 $1000$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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
$(1+6+12+18+\cdots +1200)A=(1+6(1+2+3...+200))A$ ,
where $A$ is the area of one block. Then, because $n(n+1)/2$ is equal to the sum of the first $n$ integers:
$(1+6(1+2+3...+200))=(1+6((200)(201)/2))A=120601A$ .
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 $\boxed{803}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.