2020 IMO Shortlist A5

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2 n$ real numbers $x_{1}<\ldots<x_{2 n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2 n$ values $P\left(x_{1}\right), \ldots, P\left(x_{2 n}\right)$, and writes down these $2 n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick? (Luxembourg)

魔术师打算表演以下魔术。她向观众宣布一个正整数 $n$ 以及 $2 n$ 实数 $x_{1}<\ldots<x_{2 n}$。然后，一名观众秘密选择一个实数系数为 $n$ 的多项式 $P(x)$，计算 $P\left(x_{1}\right), \ldots, P\left(x_{2 n}\right)$ 的 $2 n$ 值，并以非递减顺序将这 $2 n$ 值写在黑板上。之后魔术师向观众宣布秘密多项式。魔术师能找到表演这种魔术的策略吗？ (卢森堡)