MathDB

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})

(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.

Problem Statement