Difference between revisions of "2002 AIME II Problems/Problem 4"

Revision as of 16:08, 21 April 2008

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

$(1+6+12+18+\cdots +1200)A$ ,

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

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

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

@@ Line 7: / Line 7: @@
 == Solution ==
-{{solution}}
+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</cmath>,
+where <math>A</math> is the area of one block. 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 Remainder \boxed{803}</math>.
 == See also ==
 {{AIME box|year=2002|n=II|num-b=3|num-a=5}}

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Difference between revisions of "2002 AIME II Problems/Problem 4"

Revision as of 16:08, 21 April 2008

Problem

Solution

See also