Suppose n > 1 people are positioned in a feld, so that each has a unique nearest neighbour. Suppose further that each person has a ball that is thrown at the nearest neighbour. A survivor is a person that is not hit by a ball. Prove that if n is odd, then there is at least one person surviving.
The Attempt at a Solution
I try to prove by mathematical induction.
When n = 3
Let P1, P2, P3 be the 3 people and d12, d23, d13 be the distance between P1 and P2, P2 and P3, P1 and P3 respectively
Then I consider the different cases such as d12 < d23 and d13 > d12........etc
Under each situation, I can successfully prove that there is one person surviving.
After that, I assume the statement is true for n = 2k + 1
However, I have no idea of how to continue the proof