Question 57
1. Do there exist numbers $a_{0}, \ldots, a_{2020}$, each−1 or 1, such that $a_{0} \times a_{1}+a_{1} \times a_{2}+\cdots+a_{2019} \times a_{2020}+a_{2020} \times a_{0}=1010$?
2. Do there exist numbers $a_{1}, \ldots, a_{2020}$, each−1 or 1, such that $a_{1} \times a_{2}+a_{2} \times a_{3}+\cdots+a_{2019} \times a_{2020}+a_{2020} \times a_{1}=1010$?
2. Do there exist numbers $a_{1}, \ldots, a_{2020}$, each−1 or 1, such that $a_{1} \times a_{2}+a_{2} \times a_{3}+\cdots+a_{2019} \times a_{2020}+a_{2020} \times a_{1}=1010$?