depending of course on what, exactly, you mean by "distance running," the answer would be slightly different today if you were not restricted just to the Olympics.
including, both the Olympic Games, World and European Championships, indoors and out, and the Commonwealth games, the numbers would look like this:
10,000m
Kenya 38 (8G, 16S, 14B)
Ethiopia 32 (14G, 8S, 10B)
Great Britain 32 (12G, 10S, 10B)
Finland 24 (12G, 6S, 6B)
Italy 15 (5G, 6S, 4B)
Russia 10 (3G, 1S, 6B)
United States 3 (1G, 2S, 0B)
5000m
Great Britain 52 (21G, 13S, 18B)
Kenya 42 (15G, 13S, 14B)
Finland 19 (9G, 4S, 6B)
Ethiopia 18 (5G, 7S, 6B)
Marocco 10 (4G, 3S, 3B)
United States 8 (2G, 4S, 2B)
3000m (not chase)
Spain 23 (5G, 9S, 9B)
West Germany 18 (7G, 4S, 7B)
Great Britain 15 (7G, 3S, 5B)
Russia 12 (2G, 3S, 7B)
Kenya 11 (2G, 4S, 5B)
Ethiopia 10 (6G, 0S, 4B)
United States 6 (2G, 3S, 1B)
cheers