USAJMO 2020 Question 6

Let $n\ge2$, and let $P(x_1,\ldots,x_n)$ be a nonconstant polynomial with real coefficients. Suppose $P$ vanishes whenever two of its arguments are equal. Prove that $P$ has at least $n!$ terms.

Proposed by Ankan Bhattacharya. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p15952921. Source video https://youtu.be/r7j0oRtpErA?t=4905.