Difference between revisions of "2002 AIME II Problems/Problem 4"
Greenpizza96 (talk | contribs) m |
Phoenixfire (talk | contribs) (→Solution) |
||
Line 6: | Line 6: | ||
If <math>n=202</math>, then the area of the garden enclosed by the path, not including the path itself, is <math>m\left(\sqrt3/2\right)</math> square units, where <math>m</math> is a positive integer. Find the remainder when <math>m</math> is divided by <math>1000</math>. | If <math>n=202</math>, then the area of the garden enclosed by the path, not including the path itself, is <math>m\left(\sqrt3/2\right)</math> square units, where <math>m</math> is a positive integer. Find the remainder when <math>m</math> is divided by <math>1000</math>. | ||
− | == Solution == | + | == Solution 1== |
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 | 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 | ||
Line 20: | Line 20: | ||
<math>m=361803</math>, <math>\dfrac{m}{1000}=361</math> Remainder <math>\boxed{803}</math>. | <math>m=361803</math>, <math>\dfrac{m}{1000}=361</math> Remainder <math>\boxed{803}</math>. | ||
+ | |||
+ | == Solution 2== | ||
+ | Note that this is just the definition for a centered hexagonal number, and directly applying the formula for <math>(n-1)^{th}</math> term is <math>3n(n+1)+1</math>. Applying this for <math>200</math> as we want the inner area gives <math>120601</math>. Then continue as above. | ||
== See also == | == See also == |
Revision as of 07:28, 13 September 2020
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.