A circle of radius $r$ is surrounded by three circles, whose radii are 1, 2, and 3, all externally tangent to the inner circle and externally tangent to each other, as shown in the diagram below.
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{6}{23}\qquad\textbf{(C) }\frac{3}{11}\qquad\textbf{(D) }\frac{5}{17}\qquad\textbf{(E) }\frac{3}{10}$
Difficulty: 68/100 — A direct application of Descartes Kissing Circles, but difficult without that knowledge.
Core Concepts: circles, tangent circles
Challenges: It is unclear what to do after you connect the centers of the circles, especially if looking for a clean solution.
Techniques: Whenever you have tangent circles, always connect their centers.
Error-prone Steps: Forgetting to take the reciprocal in Descartes Kissing Circle Theorem or falling into a trigonometric dead end.
Ideal Time: Experienced: ≤ 1.5 min or ≤8 min, depending on solution method
Intermediate: ≤ N/A
Beginner: ≤ N/A
Comment: The amount of time this problem takes is entirely dependent on your previous knowledge.
