Tessellate S.T.E.M.S. (2019) - Mathematics - Category A (School) - Set 5 - Objective Problem 4
In the least morbid way possible imagine an odd number of gunmen are situated on a plane surface such that all distances between them are mutually different (paintball, if you will). On a signal every gunman shoots the person nearest to him. Consider the following situations:
I. At least one person survives. II. The paths of the bullets do not cross. III. At least one person is hit by at least bullets. IV. The set of segments formed by the trajectories of the bullets may contain a closed polygon.
Determine, in sequence which of the statements are always True (T), always False (F) or may be true (M) (that is, true for a certain initial number of gunmen but not for others).
This problem is a part of Tessellate S.T.E.M.S. (2019)
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1 solution
Statement (IV) -- the set of segments formed may contain a closed polygon -- must be false.
Assume, for contradiction, that there are n gunment whose bullet paths form a closed n -gon. Since all distances between them are distinct, among these n gunmen, there must be two who are closest two each-other. Therefore they shoot each other and their bullet trajectories coincide, forming one side of the n -gon. The remaining n − 2 gunmen shoot n − 2 bullets, forming at most n − 2 bullet trajectories. But it is impossible to form a n -gon with only n − 1 segments, a contradiction.
The only answer choice for which statement (IV) is false is TTFF .