Funahashi's theorem shows that neural networks (and thus AI) are universal approximators. In basic terms, it means that any problem that is solvable can be solved by AI, at least in theory.
Whether AI can do things humans can't in practice is a different question. I would say yes, but only because of scale. Even if AI only ever gets to the intelligence of an 8 year old, a team of the world's experts might not be able to solve problems that a billion children could solve before dying of old age.
In regards to the person asking about how you could verify that the answer is correct if you don't know the question in the first place, this is because some problems are easier to verify than to solve. For example, RSA encryption which the web runs on is based on the fact that verifying the product of two large primes is much easier than finding the primes given the product.
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