Question 7
Alex and Bibi play the following game. Alex chooses a natural number $k$ not exceeding $1000$. Then Bibi chooses a collection $B$ of $n$ integers in $\{0,1,...,1000\}$, not necessarily distinct, where $n \gt k$. Now Alex is allowed to apply repeatedly the following operation on $B$: choosing $k$ numbers $b_1, ..., b_k$ from $B$ and changing them as follows. For each $i = 1, ..., k$ the number $b_i$ is replaced by $b_i + 1$ if $b_i \lt 1000$ and by $0$ if $b_i = 1000$.
Alex wins if via several operations he succeeds in making all numbers in $B$ equal to $0$; if he fails then Bibi wins. Find all $k$ that guarantee Alex a win, regardless of the collection $B$ chosen by Bibi.
Alex wins if via several operations he succeeds in making all numbers in $B$ equal to $0$; if he fails then Bibi wins. Find all $k$ that guarantee Alex a win, regardless of the collection $B$ chosen by Bibi.