IMC 1995 Problem 5

Let $A$ and $B$ be real $n\times n$ matrices. Assume there exist $n+1$ different real numbers $t_{1},t_{2},\dots,t_{n+1}$ such that the matrices $$C_{i}=A+t_{i}B, \, i=1,2,\dots,n+1$$ are nilpotent. Show that both $A$ and $B$ are nilpotent.