1. There are 101 cells. Two players take turns writing one of the digits from 0 to 9 in these cells from left to right. If the sum of all the digits written after filling all the cells is divisible by 11, the player who moved first wins; if it is not divisible by 11, the second player wins. Which player will win with the correct play and any play of the opponent? Justify your answer.
# Solution:

The winning strategy will be for the first player. Let's describe it:

To ensure the first player wins, we need the sum of all digits to be divisible by 11 on the 101st step, meaning the remainder when divided by 11 should be "0". We can achieve this if and only if the sum of all digits after the 100th step gives any remainder when divided by 11, except "1". To guarantee this, the sum after the 99th step must give a remainder of "2". This will happen if the remainder on the 98th step is anything but "3". Then, the first player needs to ensure a remainder of "4" on the 97th step, a remainder of "6" on the 95th step, a remainder of "8" on the 93rd step, a remainder of "10" on the 91st step, a remainder of "1" on the 89th step, and so on, a remainder of "10" on the 3rd step, and a remainder of "1" on the 1st step.

Answer: the first player