Frankin B.R.

At some point on a circular island with a radius of 1 km, a treasure is buried. On the shore of the island stands a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed 500 m. In addition, the mathematician has a map of the island, on which he can record all his movements, take measurements, and perform geometric constructions. The mathematician claims that he has an algorithm to reach the treasure by walking less than 4 km. Could this be true?
Let $O$ - the center of the island, $A$ - the starting point, $w$ - a circle with center at point $O$ and radius $500 \, \text{m}, l_{1}$ and $l_{2}$ - tangents to $w$, parallel to $OA$ (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_0918ecac094e52bfe73bg-38.jpg?height=506&width=497&top_left_y=-1&top_left_x=786)

We will show that it is possible to receive a signal from the device by traveling less than 3.5 km, which leads to the answer. One needs to walk 500 m towards the center of the island, and then walk along the circumference of $w$. If we did not receive a signal while moving towards the center of the island, then the signal is not in the shaded area, and therefore it is not inside the curvilinear triangle $OBC$. Moving from point $X$ to point $Y$ along the larger arc of the circle $w$, we will definitely receive a signal, as all points of the island will be covered.

Since triangle $OBC$ is equilateral, $\angle BOA = 30^\circ$, so we will travel no more than $\frac{11}{12}\pi$ along the smaller circumference, that is, in total, no more than $\frac{11}{12}\pi + 0.5 < 3.5$ km.

## Answer

yes, it can.