AMC 10 · 2025 · #5

Grade 11 geometry-2dalgebra

Archetype Geometric Series Share └ Sub-archetype Nested-Similar-Figure Area Sum

geometric-series-infinitearea-differencesimilar-figuresquadratic-equations pattern-recognitionidentify-subproblems ↑ Prerequisites: geometric-series-infinite

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

Nested squares share a center; each square's side is k times the next square out, with 0 ＜ k ＜ 1. The rings between consecutive squares are shaded in an alternating pattern, starting with the outermost ring shaded. The total shaded area is 64% of the outer square's area. Find k.

Pick an answer.

(A)

$\frac 35$

(B)

$\frac {16}{25}$

(C)

$\frac 23$

(D)

$\frac 34$

(E)

$\frac 45$

How to solve

Strategy Look for a Pattern

The figure repeats the same shrink-by-k step forever, so the shaded area is a sum of infinitely many pieces that follow a fixed rule. If we spot the pattern in those pieces, the whole sum collapses into one geometric series we can add with a formula, turning an infinite picture into a single equation for k.

1STEP 1

Normalize and list the areas

Ratios only, so let the outer side be 1. Sides run 1, k, k², k³, …, and squaring each gives areas 1, k², k⁴, k⁶, …

sides 1, k, k², k³, … → areas 1, k², k⁴, k⁶, …

2STEP 2

Write each shaded ring

A shaded ring is one square minus the square inside it. The shaded (even) rings give (1 - k²), (k⁴ - k⁶), (k⁸ - k¹⁰), …

S = (1 - k²) + (k⁴ - k⁶) + (k⁸ - k¹⁰) + …

3STEP 3

Sum the geometric series

Each pair shares the factor (1 - k²), leaving 1 + k⁴ + k⁸ + …, a geometric series with ratio k⁴ that sums to $\frac{1}{1 - k⁴}$ .

S = (1 - k²)(1 + k⁴ + k⁸ + …) = (1 - k²)·1/(1 - k⁴)

4STEP 4

Factor and cancel

Since 1 - k⁴ = (1 - k²)(1 + k²), the (1 - k²) cancels top and bottom, leaving S = $\frac{1}{1 + k²}$ .

S = (1 - k²)/(1 - k²)(1 + k²) = 1/(1 + k²)

5STEP 5

Apply the 64% condition and solve

The shaded fraction is 64% = $\frac{16}{25}$ , so $\frac{1}{1 + k²}$ = $\frac{16}{25}$ gives 1 + k² = $\frac{25}{16}$ , then k² = $\frac{9}{16}$ and k = $\frac{3}{4}$ (positive root).

1/(1 + k²) = 16/25 → 1 + k² = 25/16 → k² = 9/16 → k = 3/4

Answer

3/4

Check k =

\frac{3}{4}

forward: k² =

\frac{9}{16}

, so 1 + k² =

\frac{25}{16}

and the shaded fraction is

\frac{1}{\frac{25}{16}}

\frac{16}{25}

= 64%, exactly the given amount. The value also fits 0 ＜ k ＜ 1, so it is a valid ratio.

💡Key takeaway

When a picture repeats the same shrinking step forever, the shaded pieces form a geometric series you can add with one formula, turning the whole picture into a single equation.

Normalize and list the areas
Write each shaded ring
Sum the geometric series
Factor and cancel
Apply the 64% condition and solve

Normalize and list the areas

Write each shaded ring

Sum the geometric series

Factor and cancel

Apply the 64% condition and solve

More like this