[ Triangle inequality (miscellaneous). ] [ Pairings and groupings; bijections ]

In Old Kalitva, 50 schoolchildren live, and in Middle Boltay - 100 schoolchildren. Where should a school be built to minimize the total distance traveled by all schoolchildren?
Let's take 50 schoolchildren from Old Kalitva and 50 from Middle Boltayev and divide them into 50 pairs, with one schoolchild from each city in each pair. For each such pair, the distance they travel to school does not depend on the location of the school (of course, it must be built somewhere on the road between the cities), but there are still 50 schoolchildren left from Middle Boltayev. Therefore, to minimize the total distance traveled by all schoolchildren, the school should be built in Middle Boltayev.

## Answer

In Middle Boltayev.