First choose any US city by entering it and choosing it in the drop-down box. Then, name any US city in each 100 mile circle/ring as the rings expand outward from your city of choice, trying to name SECOND LARGEST city in each circle. The first circle will include your city of choice as a possible answer.
PLEASE NOTE - You must type a '.' (period) after each city to submit it to prevent answering another city with a name you would be have to type to get your answer. If multiple cities have the name you submitted, only the largest one will be counted.
You accumulate points based on the population of the city you named in relation to the second largest city in the circle. A smaller city’s point total will be the ratio of its population to the largest, so if you name a city of 75K people and the largest city has 100K, you get 75 points. Naming the largest city will get you points based on the ratio of the 2nd largest city's population to the largest city's.
When you run out of time or end the quiz, you will see the current ring’s 2nd largest city at right (under ‘Answer Stats’ by your coordinates).
For example, if you choose ‘Chicago’ – you will be shown a circle of radius 100 miles centered at Chicago. Naming any city in this circle (including Chicago) will cause another ‘ring’ of radius 100 miles to appear outside that circle, however - you will want to try naming the SECOND LARGEST city you can to get more points. After completing that circle, you now need to name any US city in the new ring (but not within any previous circles/rings), again trying to name the 2nd largest city. If there are not two cities within the next ring of that distance, the ring will keep expanding by that distance until two cities fall within it.
Quiz Takers 
As a mexican, some cities seem to only accept the full official name when the most common name should also be accepted. Just off the top of my head, Rosarito, Parral, La piedad, Coacalco, Taxco. Nobody here really calls them by their full name.
I’ll add these as accepted.
There are a lot of places that could be added as well: Rodolfo Sanchez Taboada (Maneadero), Ocozocoautla, Cintalapa, Motozintla, Camargo, Jimenez, Meoqui, Aldama, Ojinaga, Ahumada, Parras, Tlapa, Chilapa, Tepeji, Tula, Jacona, Caborca, Cananea.
Only the full names are accepted and barel no one calls them by their full name
In “#25” mode I got 22 of them, then pressed the button to save those 22 to my learned list. Indeed, when I pull up the Learn utility for Mexico, I only see the 3 I missed. However, in North America the ones I got are still listed.
Yes for learning it only matches it up with the exact same quiz you took – so this quiz is only paired with the Mexico Learning one, not North America.