The first step, obviously, is to identify the countries from their flags.
Each country represented has its US embassy on Massachusetts Avenue NW in Washington (and every such country is represented at least once). Take the street numbers of each country's embassy and put it into each equation in place of that country's flag. The resulting numbers are the street numbers of other countries. The answer countries, in order:
Trinidad & Tobago
The first letters of the answer countries spell out GIVE US A TEAM FLAG. When teams gave us their team flag, they were given the answer, SYNONYMOUS.
2006 MIT Mystery Hunt