Badminton II

Algebra Level 3

Two players are playing a shortened version of badminton: a \(k\)-point, \(n\)-game match with no deuce, where \(n,k > 1\) are integers, and \(n\) is odd. Specifically, in each game, the player who first scores \(k\) points wins. The winner of the match is the player who first wins \(\lceil \frac{n}{2}\rceil\) out of \(n\) games.

Is it possible for the loser to earn possible maximum amount of points not strictly more than the winner's possible minimum amount of points? If so, how many choices of \(n\) and \(k\) are there?


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