2025 AMC 10A Problem 13

Problem 13 of 25IntermediateAlgebra 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 side be

1.

The squares have sides

1, k, k^2, \ldots,

and the shaded rings alternate, so the shaded area is

1 - k^2 + k^4 - k^6 + \cdots = \frac{1}{1 + k^2}.

We’re told this equals

64\% = \frac{16}{25},

1 + k^2 = \frac{25}{16}.

Then

k^2 = \frac{9}{16},

giving

k = \frac{3}{4}.

Thus, D is the correct answer.

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