Maths Olympiad Prep

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Geometry Difficulty 3.0 Junior Find the answer

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?

Pick one

Solution

There are 1+3+5+...+2003=10022=10040041+3+5+...+2003=1002^{2}=1004004 small equilateral triangles.
Each small equilateral triangle needs 33 toothpicks to make it.
But, each toothpick that isn't one of the 1002⋅3=30061002\cdot3=3006 toothpicks on the outside of the large equilateral triangle is a side for 22 small equilateral triangles.
So, the number of toothpicks on the inside of the large equilateral triangle is 10040004⋅3−30062=1504503\frac{10040004\cdot3-3006}{2}=1504503
Therefore the total number of toothpicks is 1504503+3006=(C) 1,507,5091504503+3006=\boxed{\mathrm{(C)}\ 1,507,509}
~dolphin7

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.