2023 Putnam A2

Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.

令 $n$ 为偶正整数。令 $p$ 为 $2n$ 次的模数实多项式；也就是说，对于一些实数系数 $a_0, \dots, a_{2n-1}$，$p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$。假设对于所有整数 $k$，$p(1/k) = k^2$ 使得 $1 \leq |k| \leq n$。查找 $p(1/x) = x^2$ 的所有其他实数 $x$。