IZhO 2019 Problem 6

On a polynomial of degree three it is allowed to perform the following two operations arbitrarily many times:

(i) reverse the order of its coefficients including zeroes (for instance, from the polynomial $x^{3}-2 x^{2}-3$ we can obtain $-3 x^{3}-2 x+1$;

(ii) change polynomial $P(x)$ to the polynomial $P(x+1)$.

Is it possible to obtain the polynomial $x^{3}-3 x^{2}+3 x-3$ from the polynomial $x^{3}-2$ ?

The answer is no.