Problem 12
← Back to AMC 10B 2025
The figure below shows an equilateral triangle, a rhombus with a $60^\circ$ angle, and a regular hexagon, each of them containing some mutually tangent congruent disks. Let $T, R,$ and $H,$ respectively, denote the ratio in each case of the total area of the disks to the area of the encolsing polygon. Which of the following is true?
$$\textbf{(A)}\ T=H=R \qquad \textbf{(B)}\ H\lt R=T \qquad \textbf{(C)}\ H=R\lt T \qquad \textbf{(D)}\ H \lt R \lt T \qquad \textbf{(E)}\ H \lt T \lt R \qquad $$
Difficulty: 24/100 - Geometry problem that is trivialized by one realization.
Core Concepts: circles, equilateral triangles, auxiliary lines
Challenges: The problem encourages less experienced contestants to bash the ratios for all three figures which increases its difficulty.
Techniques: By dividing the rhombus and hexagon into equilateral triangles, simple observation shows the ratio comparisons.
Error-prone Steps: N/A
Ideal Time:
Experienced: ≤ 30 sec
Intermediate: ≤ 1 min
Beginner: ≤ 2-4 min