When I wanted to do a PhD in the US, I searched for open problems on professors' personal websites and tried to solve them. I found one that seemed doable and managed to solve it (with help from a Spanish professor I was working for at the time).

I sent the professor an email with the solution. He ignored it until, 4 months later, someone asked the same question on the CS Theory Stack Exchange, which jogged his memory about my email.

He became my PhD advisor.