Question 82
Let $k$ be a positive integer and $a_{1}, a_{2}, \ldots, a_{k}$ be nonnegative real numbers. Initially, there is a sequence of $n \geq k$ zeros written on a blackboard. At each step, Nicole chooses $k$ consecutive numbers written on the blackboard and increases the first number by $a_{1}$, the second one by $a_{2}$, and so on, until she increases the $k$-th one by $a_{k}$. After a positive number of steps, Nicole managed to make all the numbers on the blackboard equal. Prove that all the nonzero numbers among $a_{1}, a_{2}, \ldots, a_{k}$ are equal.