Competition problem — Geometry — Maths Olympiad Prep

Library / /31 of 94

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

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.