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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