Question

cho a^2018+b^2018+c^2018=a^1009b^1009+b^1009c^1009+c^1009a^1009

tính A=(a-b)^2019+(b-c)^2020+(c-a)^2020

Answer 1

Ta có a^2018 + b^2018 +c^2108 = a^1009b^1009 + b^1009c^1009 +c^1009a^1009

       => a^2018 + b^2018 +c^2018 -a^1009b^1009 -b^1009c^1009 -c^1009a^1009 =0

      => 2( a^2018 +b^2108 +c^2018 -a^1009b^1009 -b^1009c^1009 -c^1009a^1009) =0

      => [(a^1009)^2 -2a^1009b^1009 +(b^1009)^2] + [(b^1009)^2 -2b^1009c^1009 +(c^1009)^2] +[(c^1009)^2 -2c^1009a^1009 +(c^1009)^2] =0

     => (a^1009 -b^1009)^2 + (b^1009 -c^1009)^2 + (c^1009 -a^1009)^2 =0

   Vì (a^1009 -b^1009)^2 , (b^1009-c^1009)^2 , (c^1009- a^1009)^2 >_0 ( với mọi a,b,c)

    => a^1009 -b^1009 =0 , b^1009-c^1009 =0 , c^1009-a^1009 =0

   => a=b=c=0

 Thay vào A : A=0

Vậy A=0