Problem #PRU-65751

Problem

A cubic polynomial $f(x)$ is given. Let’s find a group of three different numbers $(a,b,c)$ such that $f(a)=b$, $f(b)=c$ and $f(c)=a$. It is known that there were eight such groups $[a_i,b_i,c_i]$, $i=1,2,\dots ,8$, which contains 24 different numbers. Prove that among eight numbers of the form $a_i+b_i+c_i$ at least three are different.