The Colombian border, mostly the Putumayo river, is very interesting. Peru comes within 5km of the equator, but the river doesn't quite get there along the border. The border breaks with the river just as it turns north and crosses the equator. There's a strange piece of Ecuador that prevents Peru from having any territory on the equator.
(The source of the river is north of the equator, it doesn't really "turn" north. More correct would be to say the Peruvian border begins following the river just after it has crossed the equator.)
Glad you liked the quiz :)
Interestingly, the first obviously false one you mention is the one most people get wrong.