Difference between revisions of "2020 AMC 10B Problems/Problem 16"
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==Solution== | ==Solution== | ||
| + | Notice that to use the optimal strategy to win the game, Bela can select the middle number in the range <math>[0, n]</math> 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 <math>\boxed{\textbf{A}}</math>. | ||
==Video Solution== | ==Video Solution== | ||
Revision as of 18:05, 7 February 2020
Contents
Problem
Bela and Jenn play the following game on the closed interval ![$[0, n]$](http://latex.artofproblemsolving.com/a/5/8/a585433224883cbcd304f34129a152b97d5ed96e.png)
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?
Solution
Notice that to use the optimal strategy to win the game, Bela can select the middle number in the range 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 
.
Video Solution
~IceMatrix
See Also
| 2020 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
