Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:

Ten different points are marked on a circle. Two players $A$ and $B$ play the following game. $A$ moves first and the players alternate their moves. In each of the moves a player connects two of the points with a straight line segment. A player whose segment crosses a segment previously drawn will lose the game. Which player has a winning strategy and what is the strategy.

Solution

Solution:

Notice that a player can draw a line segment if and only if the $10$ -gon is not partitioned into triangles. Since there is a total of $17$ segments for any partition of $10$ -gon into triangles, the first player will win the game no matter how he plays.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.