Question 28
Let $n \ge 3$. Alice and Bob play the following game: Alice chooses $k \in \{3, 4, \dots, n\}$ and draws a $3 \times k$ table, then he fills the $k$ cells of the first row with different numbers from $\{1, 2, \dots, n\}$. Then, Bob fills on the second row some of the cells (eventually none) with distinct numbers from $\{1, 2, \dots, n\}$, and the rest of them with $0$. Finally, on each cell of the third row we write the sum of the two cells above. Show that regardless how Alice plays, Bob can guarantee that on the third row he can obtain, in some order, the terms of a non-constant arithmetical progression.