2020 AMC 12B Problem 14
Problem 14 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 first plays the midpoint This choice makes the configuration symmetric about the center of the interval.
Thereafter, whenever Jenn picks a number Bela responds with its mirror image Since Bela has already chosen Jenn’s legal move satisfies Therefore so Bela’s response is far enough from Jenn’s new point. Symmetry shows that it is also far enough from every earlier point. Thus Bela always has a move whenever Jenn does, so Jenn is the first to be stuck. Bela always wins.
Thus, the correct answer is A.