\section*{Problem 1 - V11221}

An airplane first flies \(300 \mathrm{~km}\) in the direction of South, then changes its course and flies \(300 \mathrm{~km}\) in the direction of East, and then again \(300 \mathrm{~km}\) in the direction of North. It has returned to its starting point!

Where is the starting point if the flight

a) is over the northern,

b) is over the southern hemisphere?

(Earth radius \(r=6370 \mathrm{~km}\) )

Hint: Problem b) has infinitely many solutions.
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a) The starting point was the North Pole. Any straight line starting from the North Pole leads south, and no matter how far one flies east or west, if one then flies the same distance north again, one will arrive back at the North Pole. However, this assumes that the flight eastward was along a line of latitude, not in a straight line (which in this context means along a great circle).

b) The same principle applies here as in a), namely that the flight eastward is along a line of latitude. On the southern hemisphere, returning to the starting point is only possible if the point where the flight direction changes from south to east is the same as the point where it changes from east to north. The line of latitude must therefore have a circumference of \(300 \mathrm{~km}\). Given the slight curvature of the Earth, an approximate consideration is sufficient. The line of latitude must thus be \(\frac{300 \mathrm{~km}}{2 \pi}=47.75 \mathrm{~km}\) from the South Pole.

One returns to the starting point with the above route precisely when it is \(347.75 \mathrm{~km}\) from the South Pole.

All points that lie on the lines of latitude that are \(300 \mathrm{~km}\) north of the lines of latitude with a circumference of \(150 \mathrm{~km}\) or \(100 \mathrm{~km}\) or \(75 \mathrm{~km}\) also have the required property.