一道高中不等式证明题.已知实数XYZ分别不等于1,并且XYZ=1证明X^2/(x-1)^2+Y^2/(y-1)^2+Z^2/(Z-1)^2>=1

Question

一道高中不等式证明题.
已知实数XYZ分别不等于1,并且XYZ=1
证明X^2/(x-1)^2+Y^2/(y-1)^2+Z^2/(Z-1)^2>=1

Answer 1

设a=X/(x-1),b=Y/(y-1),c=Z/(Z-1) 得x=a/(a-1),y=b/(b-1),z=c/(c-1).代入xyz=1,得abc/(a-1)(b-1)(c-1)=1 化简后得到a+b+c+1=ab+bc+ac 式① 原命题可化为a^2+b^2+c^2-1>=0 a^2+b^2+c^2-1等价变形得 (a+b+c)^2-2(ab+bc...