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Original Message (ID '96160') By Bill Simpson:
Opening up a brand new fresh notebook and pasting in your post gives in a fraction of a second In[1]:= Solve[{ (a1-a2)-(f1-f2)*116992^4==0, 4*(a1-a2)+(b1-b2)*116992==0, 4*(a1-a2)+(d1-d2)*116992^3==0, 6*(a1-a2)-(c1-c2)*116992^2==0, (a2-a3)-(f2-f3)*187010^4==0, 4*(a2-a3)+(b2-b3)*187010==0, 4*(a2-a3)+(d2-d3)*187010^3==0, 6*(a2-a3)-(c2-c3)*187010^2==0, 20000==a1+b1+c1+d1+f1, 50000==a1+b1*116992+c1*116992^2+d1*116992^3+f1*116992^4, 100000==a3+b3*187010+c3*187010^2+d3*187010^3+f3*187010^4, 500000==a3+b3*254808+c3*254808^2+d3*254808^3+f3*254808^4, 28631==1/116992*(a1*(116992-1)+b1/2*(116992^2-1^2)+c1/3*(116992^3-1^3)+d1/4*(116992^4-1^4)+f1/5*(116992^5-1^5)), 61248==1/(187010-116992)*(a2*(187010-116992)+b2/2*(187010^2-116992^2)+c2/3*(187010^3-116992^3)+d2/4*(187010^4-116992^4)+f2/5*(187010^5-116992^5)), 175924==1/(254808-187010)*(a3*(254808-187010)+b3/2*(254808^2-187010^2)+c3/3*(254808^3-187010^3)+d3/4*(254808^4-187010^4)+f3/5*(254808^5-187010^5))},{a1,a2,a3,b1,b2,b3,c1,c2,c3,d1,d2,d3,f1,f2,f3}] Out[1]= {{a1 -> 384264998617354569953552836167342990841053369276823360290160/19209934337367091758699558713898228143853641373387384183, a2 -> -176040833964952705385178682292577921246785193736972400/114659719119767639759984567118450024662832499161, a3 -> 208782769537124558588083672449485490786009481102668678766586800/4725285620014900749474970650698614970849559742423772673, b1 -> -3448336093183566042093545320695426670450365992973209778625/998916585543088771452377053122707863480389351416143977516, b2 -> 22806024887550625050814974437313428221064502868885/458638876479070559039938268473800098651329996644, b3 -> -157892841498190649575908164299227446231271298176121754357755/170110282320536426981098943425150138950584150727255816228, c1 -> 316943162723795577452216626577990291233757392942912585/2663777561448236723873005474993887635947704937109717273376, c2 -> -688358482234363418642022857216174112309116605/1223037003944188157439835382596800263070213324384, c3 -> 9908459955696882223775935402004110045914696719865412345/1360882258564291415848791547401201111604673205818046529824, d1 -> -1164667503376069957865495232831348660445944114115/998916585543088771452377053122707863480389351416143977516, d2 -> 1247172572446427379017775361334080201055/458638876479070559039938268473800098651329996644, d3 -> -12882089059296331635650791508919388818659513558195/510330846961609280943296830275450416851752452181767448684, f1 -> 28679373219597181608239474623379692924257605/7991332684344710171619016424981662907843114811329151820128, f2 -> -69177618939664477961724405978645005/14676444047330257889278024591161603156842559892608, f3 -> 533466799756594413050218883202556742018821385/16330587102771496990185498568814413339256078469816558357888}} See if you get the same result when you did what I did. If so then perhaps some variable was given some value before you tried to do the solve or something else went wrong. If it still won't solve then perhaps you can attach a notebook evaluated the first time and then shows the values of all your variables and then tries the solve. Perhaps that will give a clue to help track down the problem.