SOLUTION: There are n>=3 people positioned on a field (euclidean plane)so that each has a unique neighbor. Each person has a cream pie.At a signal,everybody hurles his or her pie at the near

Question 744549: There are n>=3 people positioned on a field (euclidean plane)so that each has a unique neighbor. Each person has a cream pie.At a signal,everybody hurles his or her pie at the nearest neighbor.assuming that n is odd and that nobody can miss his or her target,true or false:there always remains at least one person not hit by a pie?Explain the method so that i can implement a program on it...

Found 2 solutions by lynnlo, ikleyn:
Answer by lynnlo(4176)   (Show Source): You can put this solution on YOUR website!

Answer by ikleyn(53107)   (Show Source): You can put this solution on YOUR website!
.
There are n>=3 people positioned on a field (Euclidean plane)so that each has a unique neighbor.
Each person has a cream pie. At a signal, everybody hurls his or her pie at the nearest neighbor.
assuming that n is odd and that nobody can miss his or her target, true or false: there always
remains at least one person not hit by a pie?
Explain the method so that i can implement a program on it...
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

According to the problem, we have a finite number of points (persons) at the plane, and each point has a unique
closest point.

So, let's take a point (a person) 'A', for which this distance to the closest neighbor is maximal (greatest) among
al other points (persons).
Since the set of points is finite, we can do it.

According to the problem, point 'A' has a unique closest point 'B', but by the choice of 'A',
for point B its closest point is positioned closer to 'B' than 'A'.

It means that 'B' will not hurl his/her pie at 'A': 'B' will hurl his/her pie to his/her closest neighbor,
which is different from 'A'.

So and thus, NOBODY will hurl his/her pie to 'A', so 'A' will remain not hit by a pie.

Thus, I proved that under given conditions, the answer is "TRUE",
and I constructed/pointed such exclusive person/point explicitly:

        it is the unique person/point A, whose distance
        from other people/points is maximal.

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