USAAAO 2022, I кръг — Задача 4

• B

Solution: We know the radius of the Earth from the formula sheet, $R_{\oplus} = 6.371 \times 10^6$ m. For the two points on the Equator, we can easily calculate the shortest distance between them as

For the points on latitude $60^{\circ}$, we can calculate the distance between them, along the latitude, as $d_{60^{\circ}} = d_{\mathrm{equator}} \times \cos 60^{\circ} = \frac{d_{\mathrm{equator}}}{2} \approx 1668\mathrm{km}$.

However, this naive approach is incorrect. The shortest distance between any two points is along the great circle connecting them, not along the latitude.

The Equator is the great circle connecting the two points on the Equator. However, the $60^{\circ}$ latitude is not a great circle!

We use spherical trigonometry to determine the arc length of the great circle connecting the points. Using the cosine formula, we know

Here, we are considering a spherical triangle with two vertices as the discussed points and third one being the pole. So, $c$ is the arc length of the great circle which we need to find. $a$ and $b$ are the arc length of points from pole which is $90^{\circ} - 60^{\circ} = 30^{\circ}$. $C$ is the angle between the two longitudes, which is $30^{\circ}$. Substituting all the quantities, we get

Arc length of great circle is

Answer: B