Questions & Answers about Он сам решает задачу.
Сам roughly corresponds to himself / by himself / on his own.
In Он сам решает задачу, сам adds emphasis:
- He is doing the solving himself, not someone else.
- He is doing it without help, not relying on others.
So the basic meaning He is solving the problem becomes more like:
- He is solving the problem himself.
- He solves the problem on his own.
Yes, Он решает задачу is a perfectly correct sentence.
Difference:
- Он решает задачу – neutral statement: He is solving a/the problem (no special emphasis).
- Он сам решает задачу – emphasizes independence or contrast:
- He, not someone else, solves it.
- He solves it without help.
So сам adds an extra shade of meaning, but is not grammatically required.