MEMO 2022 T2

Let $k$ be a positive integer and $a_1, a_2,... , a_k$ 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 $a_1, a_2, . . . , a_k$ are equal.