From the Wikipedia page on the Floor and ceiling functions:
The study of Waring's problem has led to an unsolved problem:
Are there any positive integers $k\ge6$ such that
$3^k-2^k\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor > 2^k-\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor-2$?
Mahler has proved there can only be a finite number of such $k$; none are known.
What is the name of this unsolved problem? Note that it is not called Waring's problem, but it is derived from the study of Waring's problem.
Update: Although I have not spent time researching this question further, I have found a related conjecture called Euler's conjecture.
Define $g(k)$ as the quantity appearing in Waring's problem, then Euler conjectured that
$$g(k) = 2^k + \left\lfloor(\frac{3}{2})^k\right\rfloor-2$$
where $\left\lfloor x \right\rfloor$ is the floor function.
The MathWorld article on Waring's problem is useful, but does not name this question's specific problem.
I have yet seen no name for the problem in question.