USAAAO 2024, I кръг — Задача 13

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

Samvit observes a binary star system of masses $M_1$ and $M_2$. Unfortunately, the star with mass $M_2$ is too dim for him to observe it, leading to the following snapshot below.

What could Samvit hypothesize to be the position and mass of the other star at this instant that would be consistent with the laws of physics and the orbit snapshot that he sees? To be clear, he has no knowledge of the value of $M_2$ or the period of the binary system.

(a) At $(\sqrt{a^2 - b^2}, 0)$ with mass $M_1 + M_2$

(b) At $(0, -b)$ with mass $M_1$

(c) At $(\sqrt{a^2 - b^2}, 0)$ with mass $\frac{M_1 M_2}{M_1 + M_2}$

(d) At $(-2\sqrt{a^2 - b^2}, -b)$ with mass $M_1$

(e) At $(\frac{M_2}{M_1}\sqrt{a^2 - b^2}, 0)$ with mass $M_2$

Отговори

Решение

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