已知:a^4+b^4+c^4+d^4=4abcd,且a,d,c,d都是负数,试证明a=b=c=d.RT

Question

已知:a^4+b^4+c^4+d^4= 4abcd,且a,d,c,d都是负数,试证明a=b=c=d.
RT

Answer 1

a^4+b^4+c^4+d^4=4abcd,
(a^4-2a^2b^2+b^4)+(c^4-2c^2d^2+d^4)+(2a^2b^2-4abcd+2c^2d^2)=0,
(a^2-b^2)^2+(c^2-d^2)^2+2(ab-cd)^2=0,
a^2=b^2,c^2=d^2.
同理可证,a^2=c^2,b^2=d^2,
所以,a^2=b^2=c^2=d^2,
a,b,c,d为负数,
∴a=b=c=d.