2020 AMC 10B Problems/Problem 16

Problem

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?

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

Solution

Notice that to use the optimal strategy to win the game, Bela can select the middle number in the range $[0, n]$ and can then mirror whatever number Jenn selects. Therefore, if Jenn can select a number within the range, so can Bela. Jenn will always be the first person to run out of a number to choose, so the answer is $\boxed{\textbf{(A)} \text{ Bela will always win}}$ .

Video Solution

https://youtu.be/3BvJeZU3T-M

~IceMatrix

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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2020 AMC 10B Problems/Problem 16

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Problem

Solution

Video Solution

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