USAAAO — 2024 First Round

At 6am on March 20th, as the Sun is rising, Leo, who is at (40° N, 75° W), plants a stick vertically on the ground. At that moment, he marks out a (finite) line on the ground in the direction of the shadow of the stick at that moment, labeling it with the current time. Every hour afterwards, on the hour, he marks out a new line in the current direction of the shadow, until the sun sets at 6pm. Three months later, Leo returns to the same spot, where the vertical stick and lines remain. Again, every hour on the hour, he marks out a line in the current direction of the shadow, until the sun sets. Let α12 and α6 be the azimuths of the lines drawn in the spring at 12pm and 6pm, and β12 and β6 be the azimuths of the lines drawn in the summer at 12pm and 6pm. Which of the following statements is true? Ignore atmospheric effects and the equation of time.

Leo then realizes that, in order for a single set of hour markings to accurately describe the time over the course of an entire year, the stick may need to be tilted away from the vertical position. More specifically, consider straight lines drawn on the ground from the base of the stick; the shadow at a certain fixed time of day, on different days of the year, should always lie on the same line. Measured as an angle from the vertical, how much does the stick need to be tilted, and in which direction?