1998 OIM Problems/Problem 1

Problem

There are 98 points on a circle. María and José play alternately in the following way: each of them draws a segment joining two of the given points that have not been joined together previously. The game ends when all 98 points have been used as ends of a segment at least once. The winner is the person who makes the last stroke. If José starts the game, who can ensure victory?

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

Let the last three points that are connected to any other point be $A$ , $B$ , and $C$ , with $C$ being the last point connected to any other point. Notice that if $3$ points remain unconnected, then whichever player is forced to connect one of these points will lose. As a result, the players will attempt to connect every point that is not $A$ , $B$ , or $C$ , which is ${95\choose2}=95\cdot47$ segments. Clearly José will draw the last segment of this set, so María will be forced to connect one of the final three points to some other point, after which José wins.

~ eevee9406

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1998 OIM Problems/Problem 1

Problem

Solution

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