Question 3
Find all lists $\left(x_{1}, x_{2}, \ldots, x_{2020}\right)$ of non-negative real numbers such that the following three conditions are all satisfied:
(i) $x_{1} \leq x_{2} \leq \ldots \leq x_{2020}$;
(ii) $x_{2020} \leq x_{1}+1$;
(iii) there is a permutation $\left(y_{1}, y_{2}, \ldots, y_{2020}\right)$ of $\left(x_{1}, x_{2}, \ldots, x_{2020}\right)$ such that
$$
\sum_{i=1}^{2020}\left(\left(x_{i}+1\right)\left(y_{i}+1\right)\right)^{2}=8 \sum_{i=1}^{2020} x_{i}^{3}
$$
A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example, $(2,1,2)$ is a permutation of $(1,2,2)$, and they are both permutations of $(2,2,1)$. Note that any list is a permutation of itself.
(i) $x_{1} \leq x_{2} \leq \ldots \leq x_{2020}$;
(ii) $x_{2020} \leq x_{1}+1$;
(iii) there is a permutation $\left(y_{1}, y_{2}, \ldots, y_{2020}\right)$ of $\left(x_{1}, x_{2}, \ldots, x_{2020}\right)$ such that
$$
\sum_{i=1}^{2020}\left(\left(x_{i}+1\right)\left(y_{i}+1\right)\right)^{2}=8 \sum_{i=1}^{2020} x_{i}^{3}
$$
A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example, $(2,1,2)$ is a permutation of $(1,2,2)$, and they are both permutations of $(2,2,1)$. Note that any list is a permutation of itself.