Intelligence is not as cool as you think it is.
[0] https://mathstodon.xyz/@tao/115855840223258103
[1] https://huggingface.co/blog/dlouapre/gpt-single-minus-gluons
[2] https://deepmind.google/blog/alphaevolve-a-gemini-powered-co...
Yes, and that's exactly what they do.
No, none of the problems you gave to the LLM while toying around with them are in any way novel.
Do you not consider that novel problem solving?