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Harvard-MIT Math Tournament

Geometry Difficulty 4.6 AIME Prove it United States

Problem:

Two circles have centers that are $d$ units apart, and each has diameter $\sqrt{d}$ . For any $d$ , let $A(d)$ be the area of the smallest circle that contains both of these circles. Find $\lim_{d \rightarrow \infty} \frac{A(d)}{d^{2}}$ .

Solution

Solution:

This equals $\lim_{d \rightarrow \infty} \frac{\pi\left(\frac{d+\sqrt{d}}{2}\right)^{2}}{d^{2}} = \frac{\pi}{4}$ .

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