Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Prove it Soviet Union

Problem:

The beam of a lighthouse on a small rock penetrates to a fixed distance $d$ . As the beam rotates the extremity of the beam moves with velocity $v$ . Prove that a ship with speed at most $v/8$ cannot reach the rock without being illuminated.

Solution

Solution:

Let the lighthouse be at $L$ . Take time $t = 0$ at the moment the boat starts its run, so that at $t = 0$ it is at $S$ a distance $d$ from $L$ , and thereafter it is at a distance less than $d$ . Take $A$ and $B$ a distance $d$ from $L$ so that $ALBS$ is a semicircle with diameter $AB$ and $S$ the midpoint of the arc $AB$ .

During the period to $t = 2.5\pi d / v$ the boat has traveled a distance less than $d$ , so it cannot reach $AB$ . But it is a distance less than $d$ from $L$ , so it must be inside the semicircle. But during this period the beam sweeps across from $LA$ to $LB$ and so it must illuminate the boat.

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