Prove that any positive integer n 1 is either a prime or can be represented as product of primes factors 2 Set contains all positive integers from 1 to 2 Prove that among any 1 numbers chosen from there are two numbers such that one is a factor of ID: 33072
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MathematicalInductioninAlgebra 1. Provethatanypositiveintegern1iseitheraprimeorcanberepresentedasproductofprimesfactors. 2. SetScontainsallpositiveintegersfrom1to2n.Provethatamonganyn+1numberschosenfromStherearetwonumberssuchthatoneisafactoroftheother. 3. Provethatif(x+1=x)isintegerthen(xn+1=xn)isalsointegerforanypositiveintegern. 4. ForsequenceofFibonaccinumbersu1=1;u2=1;uk+1=uk+uk1,k=2;3;:::provetheformulauk+m=uk1um+ukum+1 5. Provethefollowingidentities: (a) 12+22+32++n2=n(n+1)(2n+1)=6 (b) 13+23+33++n3=n2(n+1)2=4 (c) 123+234++n(n+1)(n+2)=n(n+1)(n+2)(n+3)=4 (d) 11!+22!++nn!=(n+1)!1 6. Provethefollowingdivisibilities: (a) 6=n3+5n (b) 7=(62n1+1) (c) 3n+1=23n+1 7. Provethefollowinginequalities: (a) 1=(n+1)+1=(n+2)+1=(n+3)++1=2n13=24(n1) (b) 1=12+1=22+1=32++1=n22 (c) (x1++xn)=n(x1xn)1=n,wherex1;:::;xnarepositivenumbers 4. Pentagonalnumbers.TodenesequenceofpentagonalnumbersPm(n)oneneedstopackballsintoregularm-gonswithnballsoneachside.Forexample,P4(1)=1,P4(2)=4,P4(3)=9,P4(4)=16,P4(5)=25.(Lookatpicturesbelow) EstablishtheformulaforPm(n),wherem3. 5. Establishandproveanestimateforminimalnumberofoperationstosolvethe\TheTowersofHanoi"puzzle(problem1,MathematicalInductioninProcesses)MathematicalInductioninProcesses 1. \TheTowersofHanoi"isapuzzlewith3nailsand7rings,allofdierentsizes.Initiallyallringsareonthesamenailindecreasingorderfromthebottomtothetop.Theprocedureisremovingthetopringfromanynailandplacingitonanothernail.Itisnotallowedtoplaceabiggerringonthetopofasmallerone.Isitpossibletoreplacealltheringsfromonenailontoanotherbyapplyingthisprocedure?Whatistheminimalnumberofoperationsneededtosolvethepuzzle?Establishandprovetheformula. 2. Eachofnidenticalinshapejarsislledwithapaintto(n1)=nofitsvolume.Notwojarscontainthesamekindofthepaint.Itisallowedtopouranyamountofpaintfromonejartoanother.Couldonegetthesamemixtureinalljars?Paintisnotdisposableandthereisnootheremptyvessels. 3. Abandofnpiratesdivideapileoftreasuresbetweenthemselves.Eachwantstobesurethathegetsnolessthan1=n{thofthepile.Findafairwaytosplitthepilebetweenthem(sothatanyonecanblamenooneexceptmaybehimself).Thepirates'opinionsonsizeofthepilescouldbedierent. 4. (TournamentofTowns2005,FallRound)Thereare1000potseachcontainingvaryingamountsofjam,notmorethan1/100-thofthetotal.Eachday,exactly100potsaretobechosen,andfromeachpot,thesameamountofjamiseaten.Provethatitispossibletoeatallthejaminanitenumberofdays.