Putnam 1979 A5

Problem A-5 Denote by $[x]$ the greatest integer less than or equal to $x$ and by $S(x)$ the sequence $[x],[2 x],[3 x], \ldots$. Prove that there are distinct real solutions $\alpha$ and $\beta$ of the equation $x^{3}-10 x^{2}+29 x-25=0$ such that infinitely many positive integers appear both in $S(\alpha)$ and in $S(\beta)$.