USAMO 1974 Problem 1

$p(x)$ is a polynomial with integral coefficients. Show that there are no solutions to the equations $p(a)=b$, $\mathrm{p}(\mathrm{b})=\mathrm{c}, \mathrm{p}(\mathrm{c})=\mathrm{a}$, with $\mathrm{a}, \mathrm{b}, \mathrm{c}$ distinct integers.