AMC 10 · 2025 · #13

Grade 8 geometry-2d
Archetype Geometric Series Share └ Sub-archetype Nested-Similar-Figure Area Sum
sequences-geometricgeometric-series-infinitearea-difference convert-to-algebra ↑ Prerequisites: sequences-geometric
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A big square holds infinitely many smaller squares, all with the same center and parallel sides. Each square's side is k times the side of the square just outside it, with 0 ＜ k ＜ 1. The frames between neighboring squares are shaded in two alternating colors. One color's frames together cover 64% of the big square's area. Find k.

Pick an answer.

(A)
$\frac 35$
(B)
$\frac {16}{25}$
(C)
$\frac 23$
(D)
$\frac 34$
(E)
$\frac 45$

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

The picture repeats forever, so the winning move is to spot the pattern in the areas: each square is a fixed fraction of the one before it. That turns the shaded region into a neat repeating sum. Name the outer area, break the shaded region into frames, notice the endless sum hides a shrunken copy of itself, and one short equation replaces the infinite list. Then the 64% clue pins down k.

1STEP 1

Give the outer square area 1

Let the outer square's side be 1, so its area is 1. Every other area is then just a fraction of the whole.

outer side=1, outer area=1²=1
2STEP 2

Each area is k squared of the last

One square inward multiplies the side by k, so the area by k squared: the areas run 1, k squared, k to the fourth, and so on.

1, k², k⁴, k⁶, … (each area=k²×the previous area)
3STEP 3

Add up the shaded frames

A frame is a square minus the one inside it, and the shaded color takes every other frame: 1 minus k squared plus k to the fourth, forever.

shaded=(1-k²)+(k⁴-k⁶)+…=1-k²+k⁴-k⁶+…
4STEP 4

Fold the endless sum into one equation

Pull k squared out of every term after the 1 and the sum S reappears: S equals 1 minus k squared S, so S is 1 over (1 plus k squared).

S=1-k²(1-k²+k⁴-…)=1-k² S → S(1+k²)=1 → S=1/(1+k²)
5STEP 5

Use the 64% clue and solve for k

S is 64%, or 16 over 25, so 1 plus k squared is 25 over 16, k squared is 9 over 16, and k is 3 over 4 — choice (D).

1/(1+k²)=64/100=16/25 → 1+k²=25/16 → k²=9/16 → k=3/4
Answer
3/4
Put k = 3/4 back in: k squared is 9/16, so 1 plus k squared is 25/16, and 1 divided by 25/16 is 16/25, which is exactly 64%. The value 3/4 sits between 0 and 1 as required, and it is one of the listed choices, so answer (D) holds up.
💡Key takeaway

When a picture repeats itself forever, find the smaller copy hiding inside and let one equation do the endless work.

  • Give the outer square area 1
  • Each area is k squared of the last
  • Add up the shaded frames
  • Fold the endless sum into one equation
  • Use the 64% clue and solve for k

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