Question 115
Numbers from $1$ to $2007$ are arbitrarily written down in strip cells. Two players by turns mark cells of this strip. That player after whose move there are such two natural numbers $m \lt n$ loses that the sum of numbers of all noted cells from $m$ to $n$ divides by $2008$. Prove that there is at least one thousand initial orders of numbers in cells, such that for any non-negative integer $i \le j$: the sum of numbers of cells with numbers $i, i+1, \dots, j$ does not divide by $2008$ at which the first player has a winning strategy.