USAAAO 2024, I кръг — Задача 19

• C

Solution:

Place the tip of the stick at the center of the celestial sphere. Consider the position of the Sun at a certain fixed time of day, on different days of the year. For the same time of day, the hour angle stays the same, but the declination of the Sun varies between $\delta = -23.5^{\circ}$ to $23.5^{\circ}$, encompassing an arc of a great circle passing through the celestial poles.

Then, the shadow of the tip of the stick must always be in the plane containing the center and the arc. As the shadows on different days form a line, and the lines on the ground pass through the base of the stick, the base of the stick must also be in this plane.

Repeating this argument for different times of day, we get that the base of the stick must be in the intersection of planes defined by arcs at different hour angles. As all these planes pass through the North and South celestial poles, the intersection is the line joining the NCP and SCP, and the stick base must be on this line.

Therefore, the stick must point (upwards) in the direction of the NCP. As the altitude of the NCP is the latitude, the NCP is $40^{\circ}$ above the horizon, or $50^{\circ}$ tilted from the zenith towards the North.

Answer: C