The Collaborative International Dictionary of English v.0.48:

Problem \Prob"lem\, n. [F. probl[`e]me, L. problema, fr. Gr. ?
   anything thrown forward, a question proposed for solution,
   fr. ? to throw or lay before; ? before, forward + ? to throw.
   Cf. Parable. ]
   [1913 Webster]
   1. A question proposed for solution; a matter stated for
      examination or proof; hence, a matter difficult of
      solution or settlement; a doubtful case; a question
      involving doubt. --Bacon.
      [1913 Webster]

   2. (Math.) Anything which is required to be done; as, in
      geometry, to bisect a line, to draw a perpendicular; or,
      in algebra, to find an unknown quantity.
      [1913 Webster]

   Note: Problem differs from theorem in this, that a problem is
         something to be done, as to bisect a triangle, to
         describe a circle, etc.; a theorem is something to be
         proved, as that all the angles of a triangle are equal
         to two right angles.
         [1913 Webster]

   Plane problem (Geom.), a problem that can be solved by the
      use of the rule and compass.

   Solid problem (Geom.), a problem requiring in its geometric
      solution the use of a conic section or higher curve.
      [1913 Webster] Problematic

WordNet (r) 3.0 (2006):

problem
    n 1: a state of difficulty that needs to be resolved; "she and
         her husband are having problems"; "it is always a job to
         contact him"; "urban problems such as traffic congestion
         and smog" [syn: problem, job]
    2: a question raised for consideration or solution; "our
       homework consisted of ten problems to solve"
    3: a source of difficulty; "one trouble after another delayed
       the job"; "what's the problem?" [syn: trouble, problem]