9.1. Do there exist such integers $x, y$ and $z$ that the equality holds: $(x-$ $y)^{3}+(y-z)^{3}+(z-x)^{3}=2021 ?$ (7 points)

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# Solution:

By expanding the brackets and simplifying, we obtain the polynomial

$-3 x^{2} y+3 x y^{2}-3 y^{2} z+3 y z^{2}-3 z^{2} x+3 z x^{2}=2021$. The left side is divisible by 3, while the right side is not. Therefore, such numbers do not exist.

Answer: do not exist.

Criteria: