Số cách xếp 3 nam và 3 nữ vào 6 ghế là 6! Cách.

Suy ra: n(Ω)=6!=720n(Ω)=6!=720

a) Ta gọi A là biến cố : “Nam, nữ ngồi xen kẽ nhau”

Ta đánh số ghế như sau:

Trường hợp 1:

+ Nam ngồi ghế số 1, 3, 5 suy ra có 3! cách xếp

+ Nữ ngồi ghế số 2, 4, 6 suy ra có 3! cách xếp

Suy ra trường hợp 1 có 3!.3! = 36 cách xếp

Trường hợp 2:

+ Nữ ngồi ghế số 1, 3, 5 suy ra có 3! cách xếp

+ Nam ngồi ghế số 2, 4, 6 suy ra có 3! cách xếp

Suy ra trường hợp 1 có 3!.3! = 36 cách xếp

Suy ra:

N(A) = 3!.3! + 3!.3! = 36 + 36 = 72 cách xếp.

Vậy P(A)=n(A)n(Ω)=72720=110=0,1P(A)=n(A)n(Ω)=72720=110=0,1

b) Gọi biến cố B: “Ba bạn nam ngồi cạnh nhau”

Xem 3 bạn nam như một phần tử N và N cùng 3 bạn nữ được xem như ngồi vào 4 ghế được đánh số như sau:

_ Số cách xếp N và 3 nữ vào 4 ghế là 4!

_ Mỗi cách hoán vị 3 nam cho nhau trong cùng một vị trí ta có thêm 3! cách xếp khác nhau.

Suy ra n(B) = 4!.3!=144

Vậy : P(B)=n(B)n(Ω)=144720=15=0,2

la

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cyllysylthreonylalanylthreonylseryllysylarginylalanylalanylprolylvalylisutamylprolylleucylglutamylvalylalanylllphenylalanylthreonylcysteinylglutamyllasparaginylvalylarginylphenylalanylglalanylglycylarginylglutamylisoleucyltyeinylserylisoleucylarginylserylseryllyglutamylisoleucylleucylarginylthreonylylglycylglutamyltyrosylthreonylcysteinyltyrosylglycylserylvalylserylcysteinyylvalylthreonylvalylprolylglycylglycylleucylleucylprolylglutamylarginyllysylutamylvalylvalylleucyllysylserylvalyllmylglutamylglutamylglutamylprolyllysylglutamyllysylvalyllysyllysylprolylalanolylprolyllysylprolylvalylglutamylglutlvalylthreonyllysylarginylglutamylargiylthreonyllysylvalylprolylglutamylisolylleucylprolylalanylprolylglutamylprolnylglutamylvalyllysylthreonylisoleucyllprolylglutamylprolylthreonylprolylisoylvalylprolylvalylvalylglycyllysyllysylyllysylglutamylglutamylalanylalanyllylysylglycylvalylprolyllysyllysylthreonlalanylglutamylarginylarginyllysylleuccylglutamyllysylprolylprolylaspartylglnyltyrosylglutaminylleucyllysylalanylvlyllysylglutamylisoleucyllysylaspartyltamylserylglutamylphenylalanylvalylglylanylglutamylcysteinylleucylvalylserylhreonylthreonyltryptophylmethionyllysyeucylarginylglutamylserylprolyllysylhilanylaspartylglycyllysylaspartylarginycylaspartylvalylglutaminylleucylserylahreonylcysteinylvalylleucylarginylleucylthreonylserylthreonylalanyllysylleucrolylvalylarginylphenylalanylvalyllysyamylvalylthreonylvalylvalyllysylglycylserylcysteinylglutamylleucylasparaginyalyltryptophylarginyllysylaspartylglycysylprolylglycylarginylisoleucylvalylpylmethionylarginylalanylleucylthreonylspartylaspartylthreonylaspartylalanylgreonylvalylglutamylasparaginylalanylasysteinylserylserylcysteinylvalyllysylvnylaspartyltryptophylleucylvalyllysylpinylhistidylvalyllysylprolyllysylglycylalanylcysteinylaspartylisoleucylalanyinylisoleucyllysyltryptophylphenylalanlisoleucylprolylalanylglutamylprolylasmylisoleucylleucylarginylaspartylglycyucyllysylisoleucyllysylasparaginylalanoleucylalanylglutamyltyrosylalanylvalyylarginyltyrosylprolylalanyllysylleucylglutamylvalylglutamylleucylleucyllysyylthreonylisoleucyltyrosylglutamyllysyylaspartylalanylglutamylisoleucylserylylglycylglutaminyltryptophyllysylleucyinylprolylserylprolylthreonylcysteinyllglycylglycyllysylarginylphenylalanyllyllysylleucylaspartylglutaminylalanylgaminylalanylleucylasparaginylalanylisoylleucylthreonylvalyllysylglutamylisolanylalanylvalylprolylleucyllysylaspartrginylarginylglutaminylalanylarginylphcylthreonylarginylglutamylalanylasparaysylglycylprolylaspartylisoleucylisolenylalanylaspartylisoleucylisoleucylalaisoleucylleucylvalylisoleucylasparaginylaspartylaspartylglutamylglycylvalyltutamylglycyllysyllysylthreonylserylalareonylglycylisoleucylarginylleucyllysyucylglutamylaspartylglutaminylthreonyleonylalanylthreonylphenylalanylvalylcylutamyllysylmethionylhistidylvalylvalyinylaspartylalanyllysylleucylhistidyltcylisoleucylserylserylglutamylglycyllymylmethionyllysylglutamylvalylthreonyllglutaminylisoleucyllysylalanylglutamiylthreonylalanylglutaminylleucyllysylvlyltyrosylphenylalanylthreonylvalyllysylalanylvalylglutamyllysylaspartylglutteinylglutamylvalylseryllysylaspartylvlalanyllysylaspartylglycylglutamylglutlysyltyrosylserylisoleucyllysylalanylaoleucylleucyllysylisoleucyllysyllysylaylglycylglutamyltyrosylvalylcysteinylatyllysylthreonyllysylalanylasparaginylinylleucylisoleucylglutamylvalylglutamylglutamylvalylphenylalanylvalylglycyllalanylglutamylisoleucylglutamylleucyltidylglycylglutaminyltryptophyllysylleylthreonylalanylserylprolylaspartylcystamylaspartylglycyllysyllysylhistidyliylasparaginylcysteinylglutaminylleucylylvalylserylphenylalanylglutaminylalanalanylalanylasparaginylleucyllysylvalyleucylphenylalanylisoleucylthreonylproylphenylalanylglutamyllysylaspartylglulcysteinylglutamylvalylserylarginylglularginyltryptophylleucyllysylglycylthreonylglycylaspartylaspartylarginylphenlaspartylglycylthreonyllysylhistidylselalanylalanylphenylalanylglutamylasparnylphenylalanylglutamylalanylglutamylaycyllysylleucylisoleucylisoleucylglutalphenylalanylleucylthreonylprolylleucysylglutamyllysylglutamylserylalanylvalleucylserylhistidylaspartylasparaginyllphenylalanyllysylasparaginylaspartylgnylthreonylarginylserylvalylserylmethillysylthreonylhistidylserylisoleucylthylserylisoleucylaspartylaspartylthreonlylglutamylalanylmethionylglycylmethiocylthreonylvalylleucylglutamylglycylasonylglycyllysylleucylglutaminylaspartylysylaspartylglutamylvalylisoleucylleuucylseryllysylalanylaspartylalanylprolysylaspartylglycyllysylglutamylisoleuclanylvalylisoleucyllysylthreonylasparteucylisoleucylleucyllysyllysylalanylleylglutaminyltyrosylthreonylcysteinylasyllysylthreonylserylglycyllysylleucylainylglutamylisoleucyllysylleucylvalylalglutamylvalylmethionylglutamylthreonylalanylglutamylthreonylglutamylisoleuceucylhistidylalanylasparaginyltryptophnylleucylleucylglutaminylthreonylprolylysylglutamylglutamylglycyllysylisoleustidylasparaginylcysteinylarginylleucyycylvalylaspartylphenylalanylglutaminyerylserylalanylhistidylleucylarginylvaglycylleucylleucylarginylprolylleucyllnylalanylglycylglutamylthreonylalanyltlglutamylleucylseryltyrosylglutamylaspyptophyltyrosylleucyllysylglycyllysyllyllysylvalylvalylprolylarginylserylgluylleucylthreonylleucylarginylaspartylvylglycylglutamylvalylglutaminylleucyltnyllysylthreonylhistidylalanylasparagiamylprolylprolylvalylglutamylphenylalalaspartylglutaminylthreonylvalylglutamlvalylleucylglutamylcysteinylglutamylvalanyllysylvalyllysyltryptophylphenylaglutamylisoleucylleucyllysylseryllysyllanylaspartylglycylarginylvalylarginylpartylcysteinylthreonylprolylglutamylalthreonylcysteinylaspartylalanyllysylaylcysteinylasparaginylleucylasparaginylglutamylphenylalanylleucylarginylprolminylvalylarginylglutamyllysylglutamyllutamylcysteinylglutamylleucylserylargalyllysyltryptophylphenylalanyllysylaslysyllysylglycyllysyllysyltyrosylasparylalanylvalylarginylisoleucylleucylvallleucylleucylaspartylaspartylglutamylalutamylvalylarginylthreonylalanylarginlthreonylvalylleucylglutamylglutamylglyllysylasparaginylleucylalanylasparagimylthreonylaspartylthreonylisoleucyllylseryllysylprolylglycylalanylglutamylvglycylaspartylglutamylglutamylisoleucyinyltyrosylglutamylisoleucylleucylthreylisoleucylleucylvalylisoleucylglutamilutamylaspartylalanylglycylasparaginyleucylprolylserylserylarginylthreonylastidylglutamylleucylalanylalanylglutamylylglutaminylasparaginylleucylglutamylyllysylalanylglutamylphenylalanylvalylutamylserylphenylalanylprolylvalylglutlaspartyllysylthreonylleucylglutamylseylvalylisoleucylalanylaspartylglycyllyllysylaspartylalanylthreonylleucylglutyltyrosylvalylvalylmethionylvalylglycynylhistidylleucylthreonylvalylisoleucylvalylvalylprolylleucyllysylaspartylthaminylglutaminylglutamylvalylvalylphenlvalylasparaginylthreonylglutamylglycyenylalanylarginylasparaginylglutamylglpartylserylseryllysyltyrosylisoleucyliylleucylvalyltyrosylthreonylleucylargistidylleucylaspartylaspartylglutaminylvalylserylleucylthreonylasparaginylhisylvalyllysylserylalanylalanylasparaginmylglutamylaspartylleucylarginylisoleupartylisoleucylglutamylthreonylmethionnylphenylalanyltryptophylcysteinyllysyaginylvalylthreonylleucyllysyltryptophtamylglutamylvalylprolylphenylalanylasyrosylarginylvalylaspartyllysyltyrosyllisoleucyllysylaspartylcysteinylglycyllycylglutamyltyrosylisoleucylvalylthreysylserylvalylalanylglutamylleucylleucrolylthreonylglutamylphenylalanylvalyltylglutaminylthreonylvalylthreonylglutnylvalylphenylalanylserylcysteinylglutylalanylasparaginylvalyllysyltryptophyinylglutamylisoleucyllysylglutamylglyclglutamyllysylaspartylglycylserylisolesoleucyllysylaspartylcysteinylarginyllnylglutamyltyrosylalanylcysteinylglycyerylarginylalanylarginylleucylphenylalrolylvalylglutamylisoleucylisoleucylarisoleucylleucylglutamylalanylprolylglyanylleucylalanylglutamylleucylasparagialylglutaminyltryptophylleucylarginylavalylvalylglutaminylglycylaspartyllysyylserylglutamylglycyllysylisoleucylhisylcysteinylaspartylisoleucyllysylprolytamyltyrosylarginylphenylalanylisoleuclanylarginylalanyllysylleucylglutamyllsoleucyllysylthreonylalanylaspartylglurtylvalylglycyllysylprolylleucylthreonspartylalanyltyrosylprolyllysylalanylgylalanyllysylglutamylasparaginylglutameonylisoleucylaspartylthreonylthreonylylphenylalanylarginylisoleucylleucylglllysylglycylarginyltyrosyllysylisoleucysylhistidylglycyllysylalanylglutamylgylleucyllysylvalylisoleucylaspartylvalaraginylleucylglutamylvalylthreonylgluycylglutamylvalylserylleucylalanyltryphreonylaspartylglycylglycylseryllysylivalylglutamylarginylarginylaspartylisotophylvalylleucylalanylthreonylaspartylglutamylphenylalanylthreonylvalylthreylglycylvalylglutamyltyrosylleucylphennylasparaginylarginylvalylglycylthreonthreonylaspartylasparaginylprolylvalylsylaspartylvalylprolylglycylprolylprolleucylthreonylaspartylvalylasparaginylleucylthreonyltryptophylglutamylprolylglycylalanylglutamylisoleucylthreonylaamylleucylarginylaspartyllysylthreonylrtylthreonylalanylmethionylthreonylvalylserylalanylthreonylvalylthreonylaspanylglutamyltyrosylserylphenylalanylargaraginylarginylisoleucylglycylvalylglyonylprolylphenylalanylvalyllysylvalylalarginylprolylserylprolylprolylvalylaspartylglutaminylthreonylglutaminylseryptophylglutamylprolylprolylleucyllysyleucylleucylglycyltyrosylisoleucylisolelglutamylglycyllysylaspartylasparaginylasparaginylmethionyllysylleucylvalylpysylvalylthreonylglycylleucylglutamyllucyltyrosylarginylvalylserylalanylglutylserylaspartylprolylserylglutamylisolylalanylaspartylaspartylalanylphenylalionylaspartylleucylserylalanylphenylalylvalylisoleucylvalylprolylasparaginylcylvalylprolylserylthreonylglycyltyrosthreonyltryptophylcysteinylphenylalanyamylthreonylglycylaspartylarginylvalylrylalanyltyrosylalanylglutamylleucylvalarginylserylaspartyllysylglycylisoleuglutamylasparaginylarginylvalyllysylthsoleucylaspartylvalylasparaginylvalylinylprolyllysylglutamylleucyllysylphenynyllysylaspartylserylvalylhistidylleucrolylaspartylaspartylaspartylglycylglyyrosylvalylvalylglutamyllysylarginylglltryptophylthreonyllysylvalylmethionylpartylleucylglutamylphenylalanylthreontaminylglycyllysylglutamyltyrosylleucynylarginylasparaginyllysylcysteinylglytyrosylvalylaspartylglutamylprolylvalyrolylalanylthreonylvalylprolylaspartylllysyltryptophylarginylaspartylarginylucylphenylalanylleucylthreonyltryptophnylaspartylglycylglycylserylarginylisolylglutamylarginylcysteinylprolylarginlvalylalanylcysteinylglycylglutamylpromethionylglutamylvalylthreonylglycylletophyltyrosylalanyltyrosylarginylvalylylglutaminylglycylalanylseryllysylprollglutamylisoleucylglutaminylalanylvalyalanylprolylglutamylisoleucylphenylalaucylalanylglycylleucylthreonylvalyllyslglutamylleucylprolylalanylthreonylvalprolyllysylisoleucylthreonyltryptophyllisoleucylleucyllysylglutaminylaspartyucylglutamylasparaginylvalylprolyllysyleucylvalylaspartylseryllysylarginylserosylisoleucylisoleucylglutamylalanylvarginylalanylthreonylalanylvalylvalylgspartyllysylprolylglycylprolylprolylalucylthreonylaspartylvalylthreonylaspareucylthreonyltryptophylasparaginylprolylglycylseryllysylisoleucylthreonylaspginylarginylalanylthreonylaspartylseryylleucylserylserylthreonylvalyllysylasyllysylalanylthreonyllysylleucylisoleurosylisoleucylphenylalanylarginylvalylionyltyrosylglycylalanylglycylglutamyllylisoleucylthreonylalanyllysyltyrosyllprolylglycylprolylprolylthreonylarginisoleucylthreonyllysylaspartylalanylvacysteinylglutamylprolylaspartylaspartyleucylthreonylglycyltyrosyltryptophylvolylaspartylthreonylaspartyllysyltryptyllysylmethionylprolylvalyllysylaspartyllysylglycylleucylthreonylasparaginylalanylarginylvalylleucylalanylglutamylglycyllysylprolylseryllysylserylthreoneucyllysylaspartylprolylisoleucylasparylglycyllysylprolylthreonylvalyllysylalvalylarginylleucylasparaginyltryptophylaspartylglycylglycylalanyllysylisolecylglutamylmethionylleucyllysylthreonytophylvalylarginylvalylalanylglutamylgtaminylhistidylleucylleucylprolylglycyminylglutamyltyrosylserylphenylalanylaginyllysylalanylglycylglutamylserylgluaspartylprolylvalylleucylcysteinylargiprolylserylprolylprolylarginyltryptophaginylisoleucylthreonyllysylasparaginyryptophylthreonylvalylprolylglutamyllysoleucylthreonylasparaginyltyrosylisoltylvalylarginylarginyllysylglycyltryptlthreonylthreonylvalyllysylaspartylthreonylprolylleucylthreonylglutamylglycylarginylvalylalanylalanylglutamylasparylserylaspartyltyrosylthreonylglutamylleucylalanyllysylaspartylthreonylphenyprolylprolyltyrosylalanylleucylalanylvarginylhistidylvalylaspartylleucyllysylasparaginylaspartylglycylglycylarginyrosylvalylisoleucylglutamyllysyllysylgrginyltryptophylvalyllysylalanylglycylrtylcysteinylasparaginylphenylalanylareucylglutamylglycylthreonylglutamylvalalylarginylalanylglutamylasparaginylglylprolylserylglutamylprolylthreonylglulutamylaspartylprolylthreonylserylprolrtylleucylhistidylvalylthreonylaspartyoleucylalanylisoleucylalanyltryptophylaginylglycylglycylserylprolylisoleucylglutamylmethionylcysteinylprolylvalylglmethionylarginylvalylasparaginylserylleucyllysylphenylalanyllysylvalylglutapartyllysylglutamyltyrosylvalylleucylaginylalanylisoleucylglycylvalylserylglerylglutamylasparaginylvalylvalylalanyyllysylprolylthreonylisoleucylaspartyltylisoleucylisoleucylvalylisoleucylgluisoleucylprolylvalylprolylphenylalanylthreonylvalylseryltryptophylhistidyllylysylalanylserylaspartylarginylleucyltartylhistidylisoleucylserylalanylhistirylvalylarginylalanylaspartylalanylglylthreonylleucylglutamylasparaginyllysynylserylisoleucylasparaginylvalyllysylcylprolylcysteinyllysylaspartylisoleucthreonyllysylserylserylcysteinyllysyllylprolylglutamylphenylalanylaspartylglucylhistidyltyrosylvalylleucylglutamylrginylarginylthreonyltyrosylisoleucylpmylasparaginyllysylleucylseryltryptophsoleucylprolylasparaginylglycylglutamyinylvalyllysylalanylvalylasparaginyllyyrosylisoleucylglutamylleucyllysylaspautaminylaspartylprolyllysylglutaminylplaspartylvalylglutamylvalylhistidylasplalanylmethionylthreonylisoleucylthreoltyrosylaspartylglycylglycylseryllysyleucylisoleucylglutamyllysylisoleucylalnyltryptophyllysylarginylcysteinylasparolylisoleucylleucylthreonyltyrosylthrlglutamylglycyllysylglutamyltyrosylglunylalanylglutamylasparaginylalanylalanlserylarginylalanylthreonylprolylprolyolylisoleucylaspartylalanylprolyllysylserylleucylglutamylvalyllysylarginylglleucylaspartylalanylserylisoleucylseryonylisoleucylthreonyltryptophylisoleucalylisoleucylvalylprolylglutamylglutamlanylprolylleucylvalylarginylarginylarinylglutamylglutamylglutamylprolylphenonylglutaminylarginylleucylserylisoleuylglycylglutamylserylglutaminylleucylaylarginylprolylaspartylhistidylglycyllvalylglutamylasparaginylaspartylhistidrolylcysteinylthreonylvalylserylvalyllolylprolylisoleucylasparaginylphenylalylisoleucylarginyllysylthreonylserylvautamylprolylprolylleucylaspartylaspartlisoleucylasparaginyltyrosylthreonyllethreonyllysylprolylaspartylserylglutamonylserylthreonylleucylarginylhistidylonyllysylleucylisoleucylglutamylglycylnylarginylvalylarginylalanylglutamylasrolylglycylprolylprolylcysteinylvalylsylaspartylprolylphenylalanylglycylprolllysylprolylisoleucylvalylglutamylasparylmethionylleucylvalyllysyltryptophyltylasparaginylglycylserylprolylisoleucylglutamyllysylarginylglutamylvalylaspphylserylarginylvalylasparaginyllysylsucyllysylalanylasparaginylvalylaspartyucylthreonyltyrosylvalylphenylalanylarparaginylalanylalanylglycylprolylglycyserylaspartylprolyllysylthreonylalanyllprolylprolylglycylprolylprolylisoleuclthreonylserylserylthreonylthreonylisohylglutamylprolylprolylalanylphenylalatamylisoleucylvalylglycyltyrosylphenyleucylvalylglycylthreonylasparaginyllysreonylglutamyllysylmethionylisoleucyllreonylvalyllysylglutamylisoleucylarginyllysylleucylarginylvalylserylalanylvatamylglycylprolylprolylglycylglutamyltlvalylalanylglutamylprolylglutaminylglylleucylaspartylvalylserylvalyllysylglcylmethionylalanylglycyllysylthreonylllylvalylthreonylglycylarginylprolylvallthreonyllysylglutamylglutamylglycylglginylvalylvalylisoleucylaspartylasparalutamylleucylisoleucylisoleucyllysylastylhistidylglycylarginyltyrosylvalylisginylserylcysteinylglycylseryllysylphealylglutamylvalylphenylalanylaspartylvpartylleucyllysylprolylvalylvalylthreocysteinylleucylleucylasparaginyltryptortylaspartylglycylglycylserylglutamylisoleucylisoleucylglutamylarginyllysylathreonyltryptophylarginylglutaminylprolarginylseryllysylcysteinylaspartylisotamylglycylglutaminylglutamyltyrosyllylalanyllysylasparaginyllysylphenylalanvalylglutamylisoleucylglycylprolylisollleucylglycylprolylprolylthreonylseryltyrosylthreonylglutamylarginylglutaminonylleucylaspartyltryptophyllysylglutacylglycylserylprolylisoleucylglutaminytamyllysylarginylarginylhistidylaspartutamylarginylvalylasparaginyllysylargieonylserylphenylalanylleucylvalylglutalhistidylglutaminylmethionyltyrosylglulanylvalylasparaginylglutamylisoleucylrylleucylprolylleucylasparaginylvalylvrtylglutamylvalylprolylprolylthreonyliylvalylarginylglycylaspartylthreonylislutamylprolylvalylhistidylisoleucylprolleucylprolylmethionylprolyllysylisolearaginylglutamylthreonylvalylisoleucylalanylleucylglutaminylisoleucylthreonyinylserylglutamylalanyllysylthreonylglylalanylvalylarginylglutamylaspartyllyylthreonylalanylserylasparaginylarginyarginylasparaginylvalylhistidylvalylglolylserylprolylprolylarginylasparaginysoleucyllysylalanylglutamylserylcysteispartylalanylprolylleucylaspartylaspareucylthreonylhistidyltyrosylvalylisolenylserylarginyllysyllysylalanylglutamyeonylasparaginylthreonylalanylvalylgluucyltryptophyllysylleucylisoleucylprollglutamylphenylalanylarginylvalylarginlglycylisoleucylserylaspartylglutamylclvalylisoleucylglutaminylaspartylprolyolylprolylglycyllysylprolyllysylvalyllylserylmethionylleucylvalylseryltryptotylasparaginylglycylglycylserylprolyliphylleucylglutamyllysylarginylglutamylptophylserylarginylvalylserylarginylalylglycylleucyllysylglycylvalylglutamylrginylleucylleucylglutamylglycylvalyllginylalanylmethionylalanylisoleucylaspglycylprolylprolylserylglutamylprolylsylglycylaspartylprolylisoleucylphenylaserylcysteinylprolylglutamylvalyllysyllisoleucylserylleucylglycyltryptophyllglycylglycylserylprolylisoleucyllysylgethionylglutaminylglutamylglutamylglycllysylarginylvalylasparaginylglutamylpreonylthreonylcysteinylglutamylcysteinlysylglutamylleucylarginyllysyltyrosyllalanylvalylasparaginylglutamylalanylgylaspartylthreonylthreonylglycylglutamrtylisoleucylglutaminylglutamylglutamyleucylaspartylisoleucylglycylalanylglucysteinyllysylalanylglycylserylglutamilanylvalylisoleucyllysylglycylarginylpryptophylglutamylphenylalanylaspartylghionyllysylaspartylglycylvalylhistidylrtylalanylglutaminylleucylglutamylthreerylvalylisoleucylisoleucylisoleucylprrylhistidylthreonylglycyllysyltyrosylsaraginyllysylalanylglycylglutaminyllyslarginylvalyllysylvalylmethionylaspartlaspartylleucyllysylvalylserylaspartylcysteinylarginylleucylseryltryptophyllaspartylglycylglycylaspartylarginylisocylglutamyllysylarginylthreonylisoleucylthreonyllysylvalylasparaginylprolylathreonylphenylalanylvalylvalylprolylasaminylglutaminyltyrosylphenylalanylphelutamylasparaginylarginylphenylalanylgylglutamylthreonylisoleucylglutaminylaaspartylprolylisoleucyltyrosylprolylprsylleucyllysylisoleucylglycylleucylisonylvalylhistidylleucylseryltryptophyllartylglycylglycylserylprolylvalylthreotamylcysteinylleucylalanyltryptophylasysyllysylglutamylalanyltryptophylarginylarginylaspartylvalylglutamylglutamyllvalylglutamylaspartylleucylvalylglutamylphenylalanylarginylvalyllysylalanylalylseryllysylprolylserylalanylthreonycysteinylglutaminylarginylprolylaspartylaspartylleucyllysylglutamylphenylalautamylglycylthreonylasparaginylvalylassoleucyllysylglycylvalylprolylphenylalptophylphenylalanyllysylalanylprolylprinyllysylglutamylprolylvalylleucyltyroaraginyllysylleucylvalylvalylaspartylacylvalylisoleucylprolylglutaminylserylglycylleucyltyrosylthreonylisoleucylthinylleucylglycylthreonylalanylseryllysaraginylvalylleucylglycylarginylprolylisoleucyllysylphenylalanylglutamylserymethionylthreonylleucylseryltryptophylylaspartylglycylglycylseryllysylisoleusoleucylglutamyllysylarginylglutamylaltryptophylvalylhistidylvalylserylseryllthreonyltyrosylthreonylisoleucylprolytidylglutamyltyrosylvalylphenylalanylaminylasparaginyllysyltyrosylglycylisolrtylserylglutamylprolylglutamylthreonyylalanylserylvalylprolylglycylalanylprlserylserylvalylthreonylarginylasparagraginyltryptophylglutamylglutamylprolylserylprolylvalylthreonylglycyltyrosylsylaspartylthreonylthreonylseryllysylaparaginylarginylaspartylprolylisoleucyglycylvalylseryltyrosyllysylvalylthreocylserylaspartyltyrosylglutaminylphenyleucylasparaginylalanylalanylglycylvalylserylaspartylprolylalanylthreonylalaanylprolylprolylglycylprolylprolylphenartyltryptophylthreonyllysylserylserylhylserylprolylprolylleucyllysylaspartylglycyltyrosylisoleucylvalylglutamyltyylglutamylglutamyltryptophylglutamyllyvalylarginylglycylthreonyllysylleucylvlutamylglycylalanylphenylalanyltyrosyllanylvalylasparaginylisoleucylalanylglycylglutamylvalylthreonylaspartylvalylrtylarginylleucylvalylserylprolylasparanylserylvalylarginylaspartylarginylisylglycylvalylisoleucylarginylisoleucylcyllysylprolylprolylprolylthreonylvalyonylasparaginylglutamylarginylthreonyllthreonylisoleucylglutamylthreonylthreethionylvalylisoleucyllysylasparaginylidylglutaminylglycylvalyltyrosylseryllutamylalanylglycylglutamylarginyllysyllaspartylvalylleucylaspartylvalylprolylylphenylalanylleucylalanylhistidylasputamylserylcysteinyllysylleucylthreonylutamylaspartylaspartylglycylglycylserltyrosylvalylisoleucylglutamyllysylarginylalanyltryptophylthreonylprolylvalyylarginylglutaminylasparaginylalanyltholeucylglutaminylglycyllysylalanyltyrosoleucylalanylalanylglutamylasparaginylprolylphenylalanylvalylglutamylthreonleucylarginylglutamylprolylisoleucylthlylglutamylaspartylleucylglutamylvalylraginylthreonylvalylthreonylleucylthreyllysyltyrosylaspartylglycylglycylserynyltyrosylvalylleucylglutamylserylargitamyllysylphenylalanylhistidyllysylvalginylleucylleucylserylarginyllysyltyrosylglutamylglycylaspartylthreonyltyroslanylvalylasparaginylisoleucylvalylglyphenylalanylcysteinylthreonyllysylprolpartylglutamylleucylalanylprolylprolyllphenylalanylarginylaspartyllysylleucylglutamylalanylphenylalanylalanylleucylycyllysylprolyllysylprolyllysylvalylsrtylglutamylalanylaspartylvalylleucylgnylhistidylisoleucyllysylthreonylthreoleucylglutamyllysylisoleucyllysylalanyyllysyltyrosylcysteinylvalylvalylvalylcylserylarginyllysylglycylphenylalanylvalylvalylaspartylhistidylprolylglycylerylphenylalanylaspartylglutamylvalyltlvalylisoleucylseryltryptophyllysylproylglycylseryllysylisoleucylthreonylasputamyllysyllysylglutamylvalylglycyllysrolylvalylthreonylserylalanylserylalanylvalylseryllysylleucylleucylglutamylghenylalanylarginylisoleucylhistidylalaglycylisoleucylserylaspartylprolylleucysylalanyllysylaspartylarginylphenylallprolylaspartylglutaminylprolylisoleucllysylaspartylserylalanylleucylvalylthlylhistidylaspartylglycylglycyllysylprosylisoleucylleucylglutamyllysylarginylarginyltryptophylalanylarginylvalylthistidylprolyltyrosylthreonyllysylphenyucylleucylglutamylglycylcysteinylglutainylvalylserylalanylglutamylasparaginyycylaspartylprolylserylprolylprolylserllysylaspartylprolylisoleucylalanyllysaginylprolylglutamylalanylisoleucylaspaginylserylvalylaspartylleucylthreonylinylhistidylaspartylglycylglycylseryllleucylvalylglutamyltyrosylglutaminyllymyltryptophylarginylarginylalanylasparlserylcysteinylprolylglutamylthreonyllleucylarginylaspartylglycylglutaminyltnylvalylleucylalanylvalylasparaginylalylprolylalanylhistidylvalylprolylglutatylarginylleucylglutamylprolylprolylglalanylasparaginylmethionylalanylarginyllysylvalylglycylaspartylthreonylleucysoleucyllysylglycylvalylprolylphenylalhyllysyllysylglutamylaspartylarginylasylarginylisoleucylaspartylvalylthreonylutamylisoleucylarginylasparaginylalancylglycylisoleucyltyrosylserylleucylthlalanylglycylseryllysylthreonylvalylseaspartyllysylprolylglycylprolylprolylarylglutamylisoleucylarginyllysylaspartyltryptophyllysylglutamylprolylleucylaylisoleucylthreonylasparaginyltyrosylvrtylvalylalanylserylalanylglutaminyltrlthreonylseryllysyllysyllysylserylhistleucylasparaginylglutamylglycylasparaganylarginylvalylalanylalanylglutamylasinylglycylprolylphenylalanylvalylglutayllysylalanylleucylaspartylprolylleucyolyllysylaspartylleucylhistidylhistidyeonylglutamylvalylserylleucylvalyltryplarginylaspartylglycylglycylserylprolyylvalylglutamyltyrosylglutaminylglutamspartyltryptophylisoleucyllysylphenylaraginylleucylglutamylcysteinylvalylvaltaminylglycyllysylthreonyltyrosylarginylglutamylasparaginylisoleucylvalylglythreonylthreonylisoleucylprolylisoleucyllysylleucylvalylprolylprolylserylvaleucylisoleucylglutamylglycylleucylvalyonylvalylarginylphenylalanylprolylalanlylprolylvalylprolylthreonylalanyllysyglycylserylglutamylisoleucyllysylthreohreonylvalylglutamylthreonylaspartylaslleucylthreonylisoleucyllysylasparaginrtylthreonylglycylglutamyltyrosylglutaaraginylalanylalanylglycylseryllysylthlthreonylvalylleucylaspartylvalylprolyolylisoleucylasparaginylisoleucylleucylhistidylmethionylthreonylisoleucylserysylaspartylaspartylglycylglycylserylpylisoleucylvalylglutamyllysylglutaminyylthreonyltryptophylglycylvalylvalylseyllysylleucyllysylisoleucylprolylhistiinylglutamyltyrosylvalylphenylalanylarraginyllysylisoleucylglycylvalylglycylonylprolylthreonylvalylalanyllysylhistlylserylprolylprolylglycyllysylprolylvhreonylglutamylasparaginylalanylalanyllleucylprolyllysylserylaspartylglycylgycyltyrosyltyrosylmethionylglutamylargcyllysyltryptophylvalylarginylvalylasplalanylaspartylleucyllysylphenylalanylosylglutamylglycylasparaginylthreonyltlylphenylalanylalanylglutamylasparaginlprolylserylprolylserylserylaspartylprginylprolylisoleucyllysylprolylprolylgylprolyllysylleucyllysylaspartyllysylspartylleucylvalyltryptophylthreonyllysycylserylprolylisoleucylleucylglycyltytaminyllysylprolylglycylthreonylalanylnylisoleucylasparaginyllysylaspartylglnylcysteinylalanylphenylalanylarginylvamylglycylasparaginylglutamyltyrosylarsylalanylalanylasparaginylisoleucylvallarginylglutamylleucylalanylglutamylselisoleucylleucylhistidylprolylprolylgllylthreonylcysteinylarginylaspartylvallglycylglutaminylthreonylisoleucylargillysylglycylarginylprolylglutamylprolylthreonyllysylglutamylglycyllysylvalylnylvalylaspartylleucylisoleucylglutamiglutamylleucylglutaminylisoleucyllysylartylhistidylglycyllysyltyrosylisoleucnylserylserylglycylhistidylalanylglutalasparaginylvalylleucylaspartylarginylylasparaginylleucyllysylvalylthreonylalasparaginylcysteinylthreonylisoleucylrolylleucylaspartylasparaginylglycylglsparaginylphenylalanylisoleucylvalylglraginylglutaminyllysylglycyltryptophyltylvalylthreonyllysylarginylleucylisolucylalanylasparaginylasparaginylglutamvalylcysteinylalanylglutamylasparaginyhreonylisoleucylglutamylthreonyllysyltsoleucylasparaginylprolylisoleucylaspaylglutamylasparaginylleucylhistidylisolthreonylphenylalanylvalyltyrosylleucylaspartyltyrosylaspartylglycylglycylsesylhistidylvalylglutamylarginylarginyltyltryptophylglutamylarginylvalylhistiutamylthreonylhistidyltyrosylmethionyllutamylasparaginylglutaminylisoleucyltlylglutaminylthreonyllysylasparaginylgartyltryptophylvalyllysylthreonylglutaamylaspartylleucylglutaminyllysylprolyylserylglycylvalylleucylthreonylvalyllleucylarginylleucylglutamylalanylglycylalanylprolylglutamylvalylalanyltryptotylalanylthreonylaspartylleucylthreonylisoleucylaspartylthreonylarginylalanylserylleucylthreonyllysylalanyllysylartyrosylvalylvalylthreonylalanylthreonyylphenylalanylvalylalanyltyrosylalanylaspartyllysylprolylglycylprolylvalylarlvalylaspartylvalylserylserylaspartylayltryptophylaspartylprolylprolylglutaminylglutamylisoleucylglutaminylasparagysylcysteinylglutamylthreonyllysylargieonyltyrosylserylalanylthreonylvalylleeonylvalylthreonyla

Ta chứng minh bổ đề:

Với x,y,z dương thì:

\(8\left(x+y+z\right)\left(xy+yz+zx\right)\le9\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

\(\Leftrightarrow x\left(y-z\right)^2+y\left(z-x\right)^2+z\left(x-y\right)^2\ge0\)(đúng)

Quay lại bài toán ta có:

\(A^{2020}=\left(\sqrt[2020]{\frac{a}{a+b}}+\sqrt[2020]{\frac{b}{b+c}}+\sqrt[2020]{\frac{c}{c+a}}\right)^{2020}\)

\(=\left(\sqrt[2020]{\frac{a\left(a+c\right)}{\left(a+b\right)\left(a+c\right)}}+\sqrt[2020]{\frac{b\left(b+a\right)}{\left(b+c\right)\left(b+a\right)}}+\sqrt[2020]{\frac{c\left(c+b\right)}{\left(c+a\right)\left(c+b\right)}}\right)^{2020}\)

\(\le\left(1+1+1\right)^{2018}.2.\left(a+b+c\right).\left(\frac{a}{\left(a+b\right)\left(a+c\right)}+\frac{b}{\left(b+c\right)\left(b+a\right)}+\frac{c}{\left(c+a\right)\left(c+b\right)}\right)\)

\(=3^{2018}.\frac{4\left(a+b+c\right)\left(ab+bc+ca\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

\(\le3^{2018}.\frac{9\left(a+b\right)\left(b+c\right)\left(c+a\right)}{2\left(a+b\right)\left(b+c\right)\left(c+a\right)}=\frac{3^{2020}}{2}\)

\(\Rightarrow A\le\frac{3}{\sqrt[2020]{2}}\)