I think there are plenty of things that can't be solved, but nothing that can't be improved.
Homelessness, there are just some people who don't fit into modern life - maybe they can't be housed, but conditions could be improved.
Poverty, there is no complete solution that won't be worse than the problem (yet) but things could be so much better than they are with the means we do have.
Pollution, there is probably no way we can live with our current technology without causing pollution but again - we could make substantial improvements with the tech we do have now.