2020 AMC 12B Problems/Problem 14

Problem 14

Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$ . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$ . 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?

$\textbf{(A) } \text{Bela will always win.}$ $\textbf{(B) } \text{Jenn will always win.}$ $\textbf{(C) } \text{Bela will win if and only if }n \text{ is odd.}$ $\textbf{(D) } \text{Jenn will win if and only if }n \text{ is odd.}$ $\textbf{(E) } \text{Jenn will win if and only if }n > 8.$

Solution

We can see that if Bela chooses $\frac{n}{2}$ , she splits the line into two halves. After this, she can simply mirror Jenn's moves, and because she now goes after Jenn, Bela will always win. Thus, our answer is $\textbf{(A) } \text{Bela will always win.}$

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2020 AMC 12B Problems/Problem 14

Problem 14

Solution