EGMO 2020 Problem 2

Find all lists $(x_1, x_2, \ldots, x_{2020})$ of non-negative real numbers such that the following three conditions are all satisfied:

• $x_1 \le x_2 \le \ldots \le x_{2020}$ ;
• $x_{2020} \le x_1 + 1$ ;
• there is a permutation $(y_1, y_2, \ldots, y_{2020})$ of $(x_1, x_2, \ldots, x_{2020})$ such that $$\sum_{i = 1}^{2020} ((x_i + 1)(y_i + 1))^2 = 8 \sum_{i = 1}^{2020} x_i^3.$$

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.