Bakayev E.B.

What is the maximum number of different integers that can be written in a row so that the sum of any 11 consecutive numbers is 100 or 101?
Evaluation. Suppose we have managed to list such distinct numbers $x_{1}, ..., x_{23}$, that the sum of any 11 consecutive numbers equals $A$ or $B$ (otherwise $x_k = x_{k+11}$). Thus, $S_k = S_{k+2}$. Since $x_{1} + S_{2} + S_{13} = S_{1} + S_{12} + x_{23}$, then $x_{1}= x_{23}$. Contradiction.

Example. Let's choose 10 natural numbers with a step of 3, and the eleventh one - complementing their sum to 100. Then the sequence $x_{1}, ... x_{11}, x_{1} + 1, x_{2} - 1, x_{3} + 1, x_{4} - 1, ..., x_{11} + 1$ will be the desired one. For example, like this: 0,3, 6,9,12,15,18,21,24,27,-35,1,2,7,8,13,14,19,20,25,26,-34.

## Answer

22 numbers.