2025 AMC 12A Problem 5

Problem 5 of 25EasierAlgebra Geometry

In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is

k,

where

0 \lt k \lt 1.

The spaces between squares are alternately shaded, as shown in the figure (which is not necessarily drawn to scale).

The area of the shaded portion of the figure is

64\%

of the area of the original square. What is

k?

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Solution

Let the outer square have area

1.

The nested squares have areas

1, k^2, k^4, \ldots,

so the ring between the

n

th and

(n+1)

th squares has area

k^{2n}(1-k^2).

The shaded rings are the alternate ones

n = 0, 2, 4, \ldots,

with total area

\begin{aligned} \sum_{j=0}^{\infty} k^{4j}(1-k^2) &= \frac{1-k^2}{1-k^4} \\ &= \frac{1}{1+k^2}. \end{aligned}

Setting

\dfrac{1}{1+k^2} = \dfrac{16}{25}

gives

1 + k^2 = \dfrac{25}{16},

k^2 = \dfrac{9}{16}

and

k = \dfrac{3}{4}.

Thus, the correct answer is D.

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