From "[An Invitation to Mathematics](https://books.google.com/books?id=9TATfteVeVYC&pg=PA10)":  
> Are there any integer solutions to $x^3 + y^3 + z^3 = 33$?

I thought this might be a good candidate since that book was meant as a bridge from competitive Mathematics to research. There are a few other examples, but I am quoting only one here due to your requirement.