Problem 24 · 2018 Math Kangaroo
Stretch
Spatial & Visual Reasoning
Two concentric circles with radii 1 and 9 form an annulus. n non-overlapping circles are drawn inside this annulus, each touching both circles of the annulus. (The diagram shows an example for n = 1.) What is the biggest possible value of n?
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Answer: C — 3
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Hint 1 of 2
Each inscribed circle has radius 4, with its centre 5 from the common centre.
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Hint 2 of 2
Compare the angle each circle takes up at the centre with the full 360°.
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Approach: fit equal circles around the ring by their central angles
- A circle touching both the radius-1 and radius-9 circles has radius \(\tfrac{9-1}{2}=4\), and its centre lies on the circle of radius \(1+4 = 5\).
- Two such neighbouring circles just touch when the half-angle \(\theta\) at the centre satisfies \(\sin\theta = \tfrac{4}{5}\), so each circle takes up about \(106^\circ\).
- Three circles use about \(318^\circ < 360^\circ\) (they fit), but four would need about \(424^\circ\) (too much).
- So the largest value is 3.
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