Question

已知a,b,c,d均为正数且a*a*a*a+b*b*b*b+c*c*c*c+d*d*d*d=4abcd.求证a=b=c=

Answer 1

a4+b4+c4+d4=4abcd

a4-2a2b2+b4+c4-2c2d2+d4=4abcd-2a2b2-2c2d2

(a2-b2)2+(c2-d2)2=-2(ab-cd)2

(a2-b2)2+2(ab-cd)2+(c2-d2)2=0

(a2-b2)2>=0,2(ab-cd)2>=0,(c2-d2)2>=0

所以,

a2=b2,ab=cd,c2=d2

a=b,ab=cd,c=d

a=b=c=d.