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2003 AMC 10A Problem 23

Problem 23 of 25HarderAlgebraProblem-Solving Techniques

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have 33 rows of small congruent equilateral triangles, with 55 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 20032003 small equilateral triangles?

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Solution

A triangle with nn rows has 2n−12n - 1 small triangles in its base row, so 2n−1=20032n - 1 = 2003 gives n=1002.n = 1002. Each row kk requires 3k3k toothpicks, so the total is 3(1+2+⋯+1002).3(1 + 2 + \cdots + 1002). This equals 3⋅1002⋅10032=1,507,509.3 \cdot \dfrac{1002 \cdot 1003}{2} = 1{,}507{,}509. Thus, the correct answer is C.
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