Question

a/(b+c) + b/(a+c) + c/(a+b)=1 求a^2/(b+c)+b^2/(a+c) + c^2/(a+b)的值

Answer 1

等式两边乘以(a+b)(a+c)(b+c),得a(a+b)(a+c)+b(b+a)(b+c)+c(c+a)(c+b)=(a+b)(a+c)(b+c)
(a^3+a^2b+a^2c+abc)+(b^3+b^2a+b^2c+abc)+(c^3+c^2a+c^2b+abc)
=a^2b+a^2c+b^2a+b^2c+c^2a+c^2b+2abc
两边抵消某些项,得a^3+b^3+c^3+abc=0,a^3+b^3+c^3=-abc
把所求式通分,分母是(a+b)(a+c)(b+c),分子是
a^2(a+b)(a+c)+b^2(b+a)(b+c)+c^2(c+a)(c+b)
=a^4+a^3b+a^3c+a^2bc+b^4+b^3a+b^3c+b^2ac+c^4+c^3a+c^3b+c^2ab
=a(a^3+b^3+c^3)+b(a^3+b^3+c^3)+c(a^3+b^3+c^3)+abc(a+b+c)
=-abc(a+b+c)+abc(a+b+c)
=0