2020 AMC 10B Problem 16
Problem 16 of 25IntermediateCounting & Probability
Bela and Jenn play the following game on the closed interval of the real number line, where is a fixed integer greater than They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
Answer choices
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Solution
Bela can first choose the midpoint After that, whenever Jenn chooses a number Bela chooses the reflected number
This reflected number is legal whenever Jenn’s move is legal: distances from previously chosen numbers are preserved by the reflection about and Jenn cannot choose because it was Bela’s first move. Therefore every Jenn move has a matching Bela response, so Jenn is the first player who can run out of legal moves.
Thus, A is the correct answer.