Question

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?

Bela will always win.

Jenn will always win.

Bela will win if and only if $n$ is odd.

Jenn will win if and only if $n$ is odd.

Bela will win if and only if n >8.

Solution:

Bela can first choose the midpoint $n/2$ . After that, whenever Jenn chooses a number $x$ , Bela chooses the reflected number $n-x$ .

This reflected number is legal whenever Jenn's move is legal: distances from previously chosen numbers are preserved by the reflection about $n/2$ , and Jenn cannot choose $n/2$ because it was Bela's first move. Therefore every Jenn move has a matching Bela response, so Jenn is the first player who can run out of legal moves.

Thus, A is the correct answer.

2020 AMC 10B Problem 16

Solution:

Problem 16 in Other Years