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2025 AMC 12B Problem 25

Problem 25 of 25HarderGeometry

Three concentric circles have radii 1,1, 2,2, 3.3. An equilateral triangle with side length ss has one vertex on each circle. What is s2?s^2?

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Solution

For the common center at distances 1,2,31, 2, 3 from the vertices of an equilateral triangle of side s,s, the identity 3(1+16+81+s4)3(1 + 16 + 81 + s^4) =(1+4+9+s2)2= (1 + 4 + 9 + s^2)^2 holds. This simplifies to 2s4−28s2+98=0,2s^4 - 28s^2 + 98 = 0, i.e. (s2−7)2=0,(s^2 - 7)^2 = 0, so s2=7.s^2 = 7. Thus, the correct answer is E.

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Concepts: equilateral triangle · transformation · circle

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.