USAAAO 2024, I кръг — Задача 12

• E

Solution:

The portion of the sky visible anywhere at the bottom of the hole is a circle centered at the zenith, with angular radius

so there is direct sunlight at the bottom of the hole whenever zenith distance of the center of the Sun is less than this radius, plus the angular radius of the Sun $\theta_{\odot}$. If the center of the Sun passes directly overhead, the Sun travels a total of $2(\rho + \theta_{\odot})$ across the sky while it is visible. The declination of the Sun is $\delta = -14^{\circ}38'$ by the previous question, so the total angular distance covered by the Sun over the course of a 24 hr day is $360^{\circ}\cos\delta$ (the Sun does not traverse a great circle when it is not on the equator). Therefore, the total time is

Answer: E