Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2003 AMC 10A Problems/Problem 23"

(→‎Solution 2)
(→‎Solution 2: fix)
Line 18: Line 18:
  
 
==Solution 2==
 
==Solution 2==
−
We see that the bottom row of <math>2003</math> small triangles is formed from <math>1002</math> upward-facing triangles and <math>1001</math> downward-facing triangles. Since each upward-facing triangle uses three distinct toothpicks, and since the total number of upward-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=1507509</math>
+
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=1507509</math>
  
 
== See Also ==
 
== See Also ==

Revision as of 19:13, 3 February 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 $3$ rows of small congruent equilateral triangles, with $5$ 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 $2003$ small equilateral triangles?

2003amc10a23.gif

$\mathrm{(A) \ } 1,004,004 \qquad \mathrm{(B) \ } 1,005,006 \qquad \mathrm{(C) \ } 1,507,509 \qquad \mathrm{(D) \ } 3,015,018 \qquad \mathrm{(E) \ } 6,021,018$

Solution 1

There are $1+3+5+...+2003=1002^{2}=1004004$ small equilateral triangles.

Each small equilateral triangle needs $3$ toothpicks to make it.

But, each toothpick that isn't one of the $1002\cdot3=3006$ toothpicks on the outside of the large equilateral triangle is a side for $2$ small equilateral triangles.

So, the number of toothpicks on the inside of the large equilateral triangle is $\frac{10040004\cdot3-3006}{2}=1504503$

Therefore the total number of toothpicks is $1504503+3006=1,507,509 \Rightarrow \mathrm{(C)}$

Solution 2

We see that the bottom row of $2003$ small triangles is formed from $1002$ downward-facing triangles and $1001$ upward-facing triangles. Since each downward-facing triangle uses three distinct toothpicks, and since the total number of downward-facing triangles is $1002+1001+...+1=\frac{1003\cdot1002}{2}=502503$, we have that the total number of toothpicks is $3\cdot 502503=1507509$

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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10A_Problems/Problem_23&oldid=36563"