USAAAO 2024, I кръг — Задача 18

USAAAO 2024 - First Round · 10 февруари 2024 г.

Problem writers

Wesley Andrade, Abhay Bestrapalli, Austin Chen, Justin Chen, Ved Gund, Hagan Hensley, Sandesh Kalantre, Evan Kim, David Lee, Joe McCarty, Arjun Patel, Sahil Pontula, Tara Rezaei, Raul Teixeira, Leo Yao, David Zhang

Условие

At 6am on March 20th, as the Sun is rising, Leo, who is at $(40^{\circ}\mathrm{N}, 75^{\circ}\mathrm{W})$, 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 $\alpha_{12}$ and $\alpha_6$ be the azimuths of the lines drawn in the spring at 12pm and 6pm, and $\beta_{12}$ and $\beta_6$ 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.

(a) $\alpha_{12} = \beta_{12}, \alpha_6 = \beta_6$

(b) $\alpha_{12} > \beta_{12}, \alpha_6 = \beta_6$

(c) $\alpha_{12} < \beta_{12}, \alpha_6 = \beta_6$

(d) $\alpha_{12} = \beta_{12}, \alpha_6 > \beta_6$

(e) $\alpha_{12} = \beta_{12}, \alpha_6 < \beta_6$

Отговори

Решение

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Оригинал в Архива: usaaao_first_exam_2024-3.pdf · официални решения: usaaao_first_exam_2024_solutions-1.pdf