Cities and roads in Mathlandia have the following properties. Each road connects two cities directly and does not intersect another road. It is possible to travel from every city to any other city via roads (not necessarily using just one road). The highest recorded temperature (an integer) in every city is the average of the highest recorded temperatures of the cities that are connected to it directly by a road. Show that the highest recorded temperature in all cities is the same.
I am not really sure where to start. I have tried using algebra, but that didn't seem to work since there are an ambiguous amount of cities. Induction also doesn't seem to work.
2026-04-01 09:31:36.1775035896
Show that the highest temperature is the same
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1
Try a proof by contradiction.
Note that if the highest recorded temperatures of all the cities were not the same, then at least one of them is different. That is, it is either higher or lower than the others.
In the first case, that city's highest recorded temperature is not lower than that of any of its neighbors and is higher than the that of at least one of its neighbors. Use the definition of an average to show that this is not possible.
The second case is handled similarly.