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ASU 421 All Soviet Union MO 1986 n cities, n-1 roads, shortest distances

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August 6, 2019
combinatoricsdistance

Problem Statement

Certain king of a certain state wants to build nn cities and n−1n-1 roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be 1,2,3,...,n(n−1)/21,2,3,...,n(n-1)/2 kilometres. Is it possible for
a) n=6n=6
b) n=1986n=1986 ?
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