Question 136
Alan draws a convex 2020-gon $\mathcal{A}=A_{1} A_{2} \cdots A_{2020}$ with vertices in clockwise order and chooses 2020 angles $\theta_{1}, \theta_{2}, \ldots, \theta_{2020} \in (0, \pi)$ in radians with sum $1010 \pi$. He then constructs isosceles triangles $\triangle A_{i} B_{i} A_{i+1}$ on the exterior of $\mathcal{A}$ with $B_{i} A_{i}=B_{i} A_{i+1}$ and $\angle A_{i} B_{i} A_{i+1}=\theta_{i}$. (Here, $A_{2021}=A_{1}$.) Finally, he erases $\mathcal{A}$ and the point $B_{1}$. He then tells Jason the angles $\theta_{1}, \theta_{2}, \ldots, \theta_{2020}$ he chose. Show that Jason can determine where $B_{1}$ was from the remaining 2019 points, i.e. show that $B_{1}$ is uniquely determined by the information Jason has.