"Write on this address" would mean the address is a physical thing for you to write on, like a writing desk, table, or other tangible writing area/setting.
"Write at this address" suggests you go to the address and write there. This is like if I go to my friend's house then write at their house.
"Write to this address" means you are writing something, possibly a letter, and the writing will go to a specific destination. For example, I have lately been writing to someone in another state so each letter I write is sent to that state.
I hope this helps you better understand how each preposition is used and which would make the most sense in your exam's question.