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Difference between revisions of "2020 AMC 12B Problems/Problem 14"

(Created page with "==Problem 14== 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...")
 
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==Solution==
 
==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>\textbf{(A) } \text{Bela will always win.}</math>
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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}}

Revision as of 20:31, 7 February 2020

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 $\boxed{\textbf{(A) } \text{Bela will always win.}}$

See Also

2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by
13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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