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UNIONE MATEMATICA ITALIANA SCUOLA NORMALE SUPERIORE DI PISA Progetto Olimpiadi di Matematica

Geometry Difficulty 5.4 AIME, harder Find the answer Italy

Problem:

Dante's house is located at point $D$ at the foot of a conical mountain with base diameter $4~\mathrm{km}$ and top at point $C$ . It is known that $D$ is at a straight-line distance of $4~\mathrm{km}$ from $C$ , and that, denoting by $P$ the point diametrically opposite to $D$ with respect to the base of the mountain, the Gate of Hell is located at $3/4$ of the segment $CP$ , closer to $P$ . What is the minimum distance Dante must travel (walking on the slopes of the mountain) in order to reach the Gate of Hell from his house?

Pick one

Solution

Solution:

The answer is (B). Let us unroll onto the plane the lateral surface of the mountain, cutting it along the segment $DC$ . We obtain a circular sector with center $C$ , radius $4~\mathrm{km}$ , that is, the length of $DC$ , and bounded by an arc of circumference of $4\pi~\mathrm{km}$ , that is, the circumference of the base of the original mountain. The unrolled figure will then be a semicircle. In particular, the point $D$ will lie at one of the endpoints of the arc bounding the semicircle, and $P$ at the midpoint of this arc, so the angle $P\widehat{C}D$ will be a right angle. Moreover $CD=4~\mathrm{km}$ , the distance between $C$ and the Gate of Hell is $3~\mathrm{km}$ , so the distance between $D$ and the Gate of Hell is $5~\mathrm{km}$ by the Pythagorean theorem.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.