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## Homework Statement

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.

## Homework Equations

## The Attempt at a Solution

I try to prove by mathematical induction.

When n = 3

Let P

_{1}, P

_{2}, P

_{3}be the 3 people and d

_{12}, d

_{23}, d

_{13}be the distance between P

_{1}and P

_{2}, P

_{2}and P

_{3}, P

_{1}and P

_{3}respectively

Then I consider the different cases such as d

_{12}< d

_{23}and d

_{13}> d

_{12}........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