Or get a ton of plants...
Even a couple, I would make sure you have enough lighting and then maybe get some java fern
Going smaller will be harder to maintain than the 20. The water changes will be even more often. :-/
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Yeah IMO getting a smaller tank would be stupid, your going to spend the money on a 10 gallon and then still have all the same issues.
Curious what are you doing for water changes? I have seen every thing from 10-80% changes, I would see you having issues trying to do 80% changes a week I would say like others have said that a 25% change every other week would be fine for 3 neons in a 20 gallon.
Couple ideas
1. Plants like have been mentions will absorb nitrates or an algae scrubber however, other stuff builds up in the water and its a good idea to still do water changes/gravel vac.
2. You could get a HUGE canister like a large Aquatop, Sunsun, Marineland C-530 or 2 Cobalt EXT's or something as long as it has a flow adjustment (you might have to point the output right at a wall to disperse the flow). Now the reason for this isn't that you can go without changing the water because you have a large filter it's because something like this has like a 2g capacity so if you emptied and filled it BOOM 10% water change.
2.5. another thing about some canisters is there is a drain at the bottom or you can use them to pump water out of your tank, still the old fashioned way of filling it up tho.
3. Python or Aqueon water changer