2020 AMC 10 B
Complete problem set with solutions and individual problem pages
Bela and Jenn play the following game on the closed interval 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?(2020 AMC 10B, Question #16)
- A.
Bela will always win.
- B.
Jenn will always win
- C.
Bela will win if and only if is odd
- D.
Jenn will win if and only if is odd.
- E.
Jenn will win if and only if .
Solution 1:
Notice that to use the optimal strategy to win the game, Bela must select the middle number in the range and 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 (A)Bela will always win.
Solution 2 (Guessing): First of all, realize that the value of should have no effect on the strategy at all. This is because they can choose real numbers, not integers, so even if is odd, for example, they can still go halfway. Similarly, there is no reason the strategy would change when .
So we are left with (A) and (B). From here it is best to try out random numbers and try to find the strategy that will let Bela win, but if you can't find it, realize that it is more likely the answer is (A) Bela will always win since Bela has the first move and thus has more control.
