已知abc是三个不全相等的正数,求证:(b+c)/a+(a+c)/b+(a+b)/c

问题描述:

已知abc是三个不全相等的正数,求证:(b+c)/a+(a+c)/b+(a+b)/c

证明:(b+c)/a+(a+c)/b+(a+b)/c =b/a+c/a+a/b+c/b+a/c+b/c=(b/a+a/b) + (c/a+a/c) +(c/b+b/c)而 当 a>0,b>0,c>0时b/a+a/b ≥ 2√b/a*a/b =2;c/a+a/c ≥ 2√c/a*a/c =2;c/b+b/c ≥ 2√c/b*b/c =2;所以原式 ≥ 6...