Solution
Solution by Anduril · EN
Let’s proceed by induction to show that in 2n+1 people, at least one survives.
For 1 person, it is obvious (assuming that he is not suicidal).
Assume that the resultat was shown for 2n-1 people, let 2n+1 people play the game.
We have a finite set of distances, which has a minimum reached for at least a pair of 2 persons a and b. Due to the minimality of the chosen distance, they shoot each other (here we use the unicity of the closest neighbour, as assumed in the problem).
If one man from the 2n-1 remaining also shoots one of those 2, they have not enough bullets to be all shot.
If not, we apply the inductive hypothesis to the independant set of 2n-1 people who act as if a and b did not exist.
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