Problem of the Week 1041

Minimize Victimhood

Given finitely many points P_i in the plane (with all interpoint distances being different), define the "victim" of P_i to be the point P_j that is closest to P_i and not equal to it. Find 10 points in the plane so that the number of victims is as small as you can get (proof of optimality not asked for, though it can be done).

Example: Putting five points around a unit circle, but a little bit outside the circle so that the distances will vary, and a point at the circle's center yields six points for which there are only two victims, the center and the point among the rest that is closest to the center.

Source: Crux Mathematicorum, Oct 2005, Iranian Proposed Problem (not used) from the 2000 Olympiad in Korea.

© Copyright 2005 Stan Wagon. Reproduced with permission.