Question 15
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of a turn there are $n \ge 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k \ge 1$ either is an even number with $k \le \frac{n}{2}$ or an odd number with $\frac{n}{2} \le k \le n$. A player wins the game if she removes the last marble from the table.
*Determine the smallest number $N \ge 100\,000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the table in the beginning.*
*Determine the smallest number $N \ge 100\,000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the table in the beginning.*