Question 106
Let $f(x)$ be a cubic polynomial. We say that a triple $(a, b, c)$ of distinct real numbers is a cycle if $f(a) = b$, $f(b) = c$, and $f(c) = a$. Assume that there exist eight cycles $(a_i, b_i, c_i)$, $i = 1, 2, \dots, 8$, containing 24 distinct real numbers. Prove that among eight sums of the form $a_i + b_i + c_i$, there are at least three different numbers. (A. S. Golovanov)