2003 AMC 10A Problems/Problem 23: Difference between revisions
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<math> \mathrm{(A) \ } 1,004,004 \qquad \mathrm{(B) \ } 1,005,006 \qquad \mathrm{(C) \ } 1,507,509 \qquad \mathrm{(D) \ } 3,015,018 \qquad \mathrm{(E) \ } 6,021,018 </math> | <math> \mathrm{(A) \ } 1,004,004 \qquad \mathrm{(B) \ } 1,005,006 \qquad \mathrm{(C) \ } 1,507,509 \qquad \mathrm{(D) \ } 3,015,018 \qquad \mathrm{(E) \ } 6,021,018 </math> | ||
== Solution 1== | ==Solution== | ||
=== Solution 1=== | |||
There are <math>1+3+5+...+2003=1002^{2}=1004004</math> small equilateral triangles. | There are <math>1+3+5+...+2003=1002^{2}=1004004</math> small equilateral triangles. | ||
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So, the number of toothpicks on the inside of the large equilateral triangle is <math>\frac{10040004\cdot3-3006}{2}=1504503</math> | So, the number of toothpicks on the inside of the large equilateral triangle is <math>\frac{10040004\cdot3-3006}{2}=1504503</math> | ||
Therefore the total number of toothpicks is <math>1504503+3006= | Therefore the total number of toothpicks is <math>1504503+3006=\boxed{\mathrm{(C)}\ 1,507,509}</math> | ||
==Solution 2== | ===Solution 2=== | ||
We see that the bottom row of <math>2003</math> small triangles is formed from <math>1002</math> downward-facing triangles and <math>1001</math> upward-facing triangles. Since each downward-facing triangle uses three distinct toothpicks, and since the total number of downward-facing triangles is <math>1002+1001+...+1=\frac{1003\cdot1002}{2}=502503</math>, we have that the total number of toothpicks is <math>3\cdot 502503= | We see that the bottom row of <math>2003</math> small triangles is formed from <math>1002</math> downward-facing triangles and <math>1001</math> upward-facing triangles. Since each downward-facing triangle uses three distinct toothpicks, and since the total number of downward-facing triangles is <math>1002+1001+...+1=\frac{1003\cdot1002}{2}=502503</math>, we have that the total number of toothpicks is <math>3\cdot 502503=\boxed{\mathrm{(C)}\ 1,507,509}</math> | ||
== See Also == | == See Also == | ||
Revision as of 13:57, 1 August 2011
Problem
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have 
rows of small congruent equilateral triangles, with 
small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of 
small equilateral triangles?
Solution
Solution 1
There are 
small equilateral triangles.
Each small equilateral triangle needs toothpicks to make it.
But, each toothpick that isn't one of the 
toothpicks on the outside of the large equilateral triangle is a side for 
small equilateral triangles.
So, the number of toothpicks on the inside of the large equilateral triangle is 
Therefore the total number of toothpicks is 
Solution 2
We see that the bottom row of small triangles is formed from 
downward-facing triangles and 
upward-facing triangles. Since each downward-facing triangle uses three distinct toothpicks, and since the total number of downward-facing triangles is 
, we have that the total number of toothpicks is 
See Also
| 2003 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
