2025 AMC 12B Problem 25

Attempt Problem 25 of the 2025 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.

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25.

Three concentric circles have radii $1, 2, 3.$ An equilateral triangle with side length $s$ has one vertex on each circle. What is $s^2?$

$6$

$\dfrac{25}{4}$

$\dfrac{13}{2}$

$\dfrac{27}{4}$

$7$

Answer: E

Concepts:equilateral triangle transformation circle

Difficulty rating: 2650

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. $(s^2 - 7)^2 = 0,$ so $s^2 = 7.$

Thus, the correct answer is E.

← Problem 24#24

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