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]

Moby Thesaurus II by Grady Ward, 1.0:

196 Moby Thesaurus words for "problem":
   Chinese puzzle, ado, aggravation, annoyance, anxiety, bad news,
   baffle, bafflement, baffling problem, basis, bedevilment,
   besetment, bewilderment, blemish, bone of contention, bore, bother,
   botheration, bothersomeness, brain twister, bug, bugaboo, bugbear,
   burden, can of worms, case, catch, catechism, chapter,
   complication, concern, confoundment, confusion, conundrum, count,
   crack, crashing bore, cross-interrogatory, cross-question,
   crossword puzzle, crux, debating point, defect, defection,
   deficiency, delinquent, demand, devilment, difficult, difficulty,
   dilemma, disadvantage, discomposure, disconcert, disconcertedness,
   disconcertion, disconcertment, disturbance, disturbed, dogging,
   downer, drag, drawback, embarrassment, enigma, enigmatic question,
   essence, evil, exasperation, facer, failing, failure, fault, faute,
   feeler, fix, flaw, floorer, focus of attention, focus of interest,
   foible, frailty, gist, great ado, harassment, harrying, head,
   headache, heading, hole, hounding, illustration, imbroglio,
   imperfection, inadequacy, inconvenience, incorrigible, infirmity,
   inquiry, interrogation, interrogative, interrogatory, intractable,
   issue, jam, jigsaw puzzle, kink, knot, knotty point, leader,
   leading question, little problem, living issue, main point,
   maladjusted, matter, matter in hand, meat, mess, mind-boggler,
   molestation, moot point, motif, motive, muddle, mystery, nonplus,
   nuisance, nut, nut to crack, obstreperous, peck of troubles,
   perplexed question, perplexity, persecution, perturbation, pest,
   pickle, plight, point, point at issue, point in question, poser,
   pother, predicament, pretty pickle, puzzle, puzzlement, puzzler,
   quandary, query, question, question at issue, question mark,
   quodlibet, refractory, riddle, rift, rubric, scrape,
   sea of troubles, shortcoming, sixty-four dollar question, snag,
   something missing, stew, sticker, stumper, subject, subject matter,
   subject of thought, substance, taint, text, theme, topic,
   tough proposition, trial, trial balloon, trouble, unassuredness,
   uncontrollable, ungovernable, unmanageable, unruly, upset,
   vexation, vexatiousness, vexed question, vulnerable place,
   weak link, weak point, weakness, why, worriment, worry