Question

a >0b>0c>0且a+b+c=1求证a分之一加b分之一加c分之一大于等于9

Answer 1

因为a+b+c=1,所以(1/a+1/b+1/c)×(a+b+c)=(1/a+1/b+1/c).把等式前面的式子分解开来就是（a+b+c)/a+(a+b+c)/b+(a+b+c)/c,（a+b+c)/a+(a+b+c)/b+(a+b+c)/c=3+(b+c)/a+(a+c)/b+(a+b)/c=3+b/a+a/b+c/a+a/c+c/b+b/c,而a/b+b/a大于等于2倍根号下a/b×b/a等于2,以此类推,3+b/a+a/b+c/a+a/c+c/b+b/c≥3+2+2+2=9,当且仅当a=b=c=1/3时取等号.