$\left.\begin{array}{l}{\left[\begin{array}{l}\text { Evenness and Oddness }\end{array}\right]} \\ {\left[\begin{array}{l}\text { Convex Polygons }\end{array}\right]} \\ {\left[\begin{array}{l}\text { Systems of Points and Segments (other) }\end{array}\right]}\end{array}\right]$

Author: Sukhov K.

Two players take turns drawing diagonals in a regular $(2 n+1)$-gon ( $n>1$ ). It is allowed to draw a diagonal if it intersects (at interior points) with an even number of previously drawn diagonals (and has not been drawn before). The player who cannot make the next move loses. Who will win with correct play?
Note that on one side of each diagonal there is an even number of vertices, and on the other side - an odd number. Therefore, each diagonal intersects an even number of other diagonals of the $(2 n+1)$-gon. Suppose at some point in the game it is impossible to make a move, then each unmarked diagonal intersects an odd number of already marked ones, and therefore, an odd number of unmarked diagonals. Such a situation is possible only

Thus, if the total number of diagonals in the polygon is odd, the first player wins, and if it is even - the second. In a $(2 n+1)$-gon, the number of diagonals is $(2 n+1)(n-1)$ (see problem $\underline{60391})$, which is odd for even $n$ and even for odd $n$.

## Answer

For odd $n$, the second player wins, and for even $n$ - the first.

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