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

For a point at distances

p, q, r

from the vertices of an equilateral triangle of side

a,

there is a symmetric relation among

p^2, q^2, r^2, a^2

Use

3(p^4 + q^4 + r^4 + a^4)

= (p^2 + q^2 + r^2 + a^2)^2

with

p, q, r = 1, 2, 3

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.

Problem 25 in other years