HMMT February

Geometry Difficulty 5.0 AIME, harder Find the answer

We are given some similar triangles. Their areas are $1^{2}, 3^{2}, 5^{2} \ldots$ , and $49^{2}$ . If the smallest triangle has a perimeter of 4, what is the sum of all the triangles' perimeters?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Because the triangles are all similar, they all have the same ratio of perimeter squared to area, or, equivalently, the same ratio of perimeter to the square root of area. Because the latter ratio is 4 for the smallest triangle, it is 4 for all the triangles, and thus their perimeters are $4 \cdot 1,4 \cdot 3,4 \cdot 5, \ldots, 4 \cdot 49$ , and the sum of these numbers is $\left[4(1+3+5+\cdots+49)=4\left(25^{2}\right)=2500\right.$ .

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