Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Find the answer Italy

Problem:

A river is crossed by two bridges $TS$ and $VM$ ; the two banks $TV$ and $SM$ are two arcs of concentric circles; the two bridges $TS$ and $VM$ are aligned with the center (see the figure). A person wants to get to $V$ starting from $T$ by choosing the shortest path between the two possible ones:

(1) following the river along the arc of circle $TV$

(2) crossing the bridge $TS$ , following the river along the other bank $(SM)$ and crossing the bridge $MV$ .

We denote by $\alpha$ the angle subtended by the two arcs of circle, by $R$ the length of $OT$ and by $r$ the length of $OS$ . Which data must the person necessarily have information about in order to make the best choice?

Pick one

Solution

Solution:

The answer is (C). Measuring the angle $\alpha$ in radians, the length of the first path is $R \alpha$ , while the length of the second is $2(R-r) + r \alpha$ . Therefore the first path is the shorter one if and only if
$R \alpha < 2(R-r) + r \alpha,$
that is if and only if
$(R-r)(\alpha - 2) < 0.$
Since $R - r > 0$ the choice depends only on $\alpha$ .

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