69th Belarusian Mathematical Olympiad

Peter and Andrey play the game on the $n\times 1$ board, making moves alternate. Peter starts, and on his turn he places «+» to any empty cell. Andrey on his turn places «-» to any empty cell. The game is finished when all cells are filled. Peter's prize equals to the greatest number $k$ such that for each $\ell$ from 1 to $k$ there are $\ell$ successive cells, the amount of pluses on which is greater than the amount of minuses. Find the maximal prize which Peter can guarantee for himself.