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| − | ==Problem 14==
| + | #REDIRECT [[2020 AMC 10B Problems/Problem 16]] |
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| − | Bela and Jenn play the following game on the closed interval <math>[0, n]</math> of the real number line, where <math>n</math> is a fixed integer greater than <math>4</math>. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval <math>[0, n]</math>. 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?
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| − | <math>\textbf{(A) } \text{Bela will always win.}</math> <math>\textbf{(B) } \text{Jenn will always win.}</math> <math>\textbf{(C) } \text{Bela will win if and only if }n \text{ is odd.}</math> <math>\textbf{(D) } \text{Jenn will win if and only if }n \text{ is odd.}</math> <math>\textbf{(E) } \text{Jenn will win if and only if }n > 8.</math>
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| − | ==Solution==
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| − | We can see that if Bela chooses <math>\frac{n}{2}</math>, 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 <math>\boxed{\textbf{(A) } \text{Bela will always win.}}</math>
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| − | ==See Also==
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| − | {{AMC12 box|year=2020|ab=B|before=13|num-a=15}}
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| − | {{MAA Notice}}
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