Is there a solution in distinct positive integers $a,b,c$ to the equation $$a^3+b^3=c^4$$? If so, construct one; if not, prove that it can't possibly exist.
Don't be too put off by the appearance of this puzzle: it's nowhere near as hard as its famous relative Fermat's Last Theorem. There should be an AHA moment when you realise what you need, and the solution is very surprising if you haven't seen the like before.
This puzzle was discussed by Adam McBride in “Mathematics: The Greatest Subject in the World,” The Mathematical Gazette, vol. 89, no. 516 [November 2005].