"The university is on this bank of the river and the academy is on the other."
Translation:Университет находится на этом берегу реки, а академия - на другом.
You should use «на берегу́», not «на бе́реге». The word «берег» is one of the few words that has a separate locative form (when speaking about the location, a special variant of the Prepositional case is used, «на берегу́»; in other cases it's «о бе́реге»).
«Стои́т» is OK.
Some examples (found them here) are в шкафу́ 'in the closet', в лесу́ 'in the forest', в плену́ 'in captivity', на берегу́ 'on the shore/bank', во рту 'in one's mouth', в боку́ 'in one's side', в раю́ 'in paradise', в носу́ 'in one's nose', на лбу 'on one's forehead', в бою́ 'in the battle', на полу́ 'on the floor', на виду́ 'in visibility (=visible)'.
Some feminine nouns (ending in -ь in nominative case) have a second prepositional case form too, but its only difference from the normal prepositional case is the place of stress. Some examples are в печи́ 'in the oven', в тиши́ 'in silence', в крови́ 'in blood', в тени́ 'in shadow', на мели́ 'aground, on the shoal'.
I don't have a full list, but I have a hunch it shouldn't be long: for most nouns, the second prepositional case form doesn't exist.
Russian is like an endless puzzle, mate. But nontheless our minds are also incredibly powerful.
Hmm. I was thinking "bank" would be "берег" in Russian, hence "...на берегу...". But just to be sure I checked the drop-down provided for "bank", and it said "банке". So, like a fool I used that instead, and got marked wrong, even though everything else in my sentence was correct. I reported it.