TITLE: Not especially famous, long-open problems which higher mathematics beginners can understand
QUESTION [16 upvotes]: This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate abstract algebra, linear algebra, real analysis, topology (etc.?) that a teacher could mention the questions almost immediately after stating very basic definitions.  
If they weren't either too famous or already solved, appropriate answers might include the Koethe Nil Conjecture, the existence of odd order finite simple groups or the continuum hypothesis.
Thanks

REPLY [3 votes]: Given a unit square, can you find any point in the same plane, either inside or outside the square, that is a rational distance from all four corners? 
Or, put another way, given a square $ABCD$ of any size, can you find a point $P$ in the same plane such that the distances $AB$, $PA$, $PB$, $PC$, and $PD$ are all integers?
 
References:

Guy, Richard K. Unsolved Problems in Number Theory, Vol. 1, Springer-Verlag,        2nd ed. 1991, 181-185.
Barbara, Roy. "The rational distance problem", Mathematical Gazette 95, March              2011, 59-61.