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Problem 585

AMC 10/12, early questions

Geometry Difficulty 3.5 Find the answer All-Soviet-Union Mathematical Olympiad · Soviet Union

A spy-plane circles point $A$ at a distance $10\mathrm{km}$ with speed $1000\mathrm{km/h}$ . A missile is fired towards the plane from $A$ at the same speed and moves so that it is always on the line between $A$ and the plane. How long does it take to hit?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Solution:
Answer: $18\pi$ sec.

Let $C$ be the position of the spy-plane at the moment the missile is fired. Let $B$ be the point a quarter of the way around the circle from $C$ (in the direction the spy-plane is moving). Then the missile moves along the semi-circle on diameter $AB$ and hits the plane at $B$ .

To see this, take a point $P$ on the quarter circle and let the line $AX$ meet the semi-circle at $Q$ . Let $O$ be the center of the semicircle. The angle $BOQ$ is twice the angle $BAQ$ , so the arc $BP$ is the same length as the arc $BQ$ . Hence also the arc $AQ$ is the same length as the arc $CP$ .

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