Question

Cho a²+b²+(a-b)²=c²+d²+(c-d)².CMR:a⁴+b⁴+(a-b)⁴=c⁴+d⁴+(c-d)⁴

Answer 1

Ta có a^2 + b^2 + (a - b)^2= c^2 + d^2 + (c - d)^2.
=> a^4+b^4+(a-b)^4+2[a^2b^2+a^2(a-b)^2+b^2(a-b)2]=

=c^4+d^4+(c-d)^4+2[c^2d^2+c^2(c-d)^2+d^2(c-d)^2

<=>a^4+b^4+(a-b)^4+2[a^2b^2+(a^2+b^2)(a-b)^2]

=c^4+d^4+(c-d)^4+2[c^2d^2+(c^2+d^2)(c-d)^2

Lại có a^2 + b^2 + (a - b)^2 = c^2 + d^2 + (c - d)^2.

=> 2(a^2+b^2-ab) =2(c^2+d^2-cd)

=>a^2+b^2-ab =c^2+d^2-cd

=>(a^2+b^2)2+a^2b^2-2ab(a^2+b^2)=(c^2+d^2)^2+c^2d^2-2cd(c^2+d^2).

=>a^2b^2+(a^2+b^2)(a^2+b^2-2ab)=c^2d^2+(c^2+d^2)(c^2+d^2-2cd)

=>a^2b^2+(a^2+b^2)(a-b)^2=c^2d^2+(c^2+d^2)(c-d)^2

Từ đó bạn sẽ có đpcm