Let $m,n,p,r$ be integers such that the equations
$$mx^3+nx^2+px+r=2021,$$
$$mx^3+nx^2+px+r=2022,$$
$$mx^3+nx^2+px+r=2023$$
have integer roots $a,b,c$, respectively. Prove that any other roots of these equations are not integers.
a) Find all integer solutions of
$$(3x^2+y^2-4y-17)^3-(2x^2+2y^2-4y-6)^3=(x^2-y^2-11)^3.$$
b) Given $5x+4y+3z+6t=20$ and real numbers $x\ge y\ge z\ge t\ge0$, find the maximum and minimum of $x+y+z+t$.
A magician places $n$ coins in a row. In each move he chooses three consecutive coins, leaves the middle coin unchanged, and flips the other two. Initially all $n$ coins are heads up. Find all values of $n$ for which all coins can be made tails up.