2025 AMC 12B Problem 25

Problem 25 of 25HarderGeometry

Three concentric circles have radii

1,

2,

3.

An equilateral triangle with side length

s

has one vertex on each circle. What is

s^2?

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Solution

For the common center at distances

1, 2, 3

from the vertices of an equilateral triangle of side

s,

the identity

3(1 + 16 + 81 + s^4)

= (1 + 4 + 9 + s^2)^2

holds. This simplifies to

2s^4 - 28s^2 + 98 = 0,

i.e.

(s^2 - 7)^2 = 0,

s^2 = 7.

Thus, the correct answer is E.

Tagged: equilateral triangle · transformation · circle