Question

已知a,b,c∈R,且a+b+c=1,求证1/a+1/b+1/c≥9

Answer 1

∵a+b+c=1,a,b,c∈R+（否则不对）

∴1/a+1/b+1/c

=(a+b+c)/a+(a+b+c)/b+(a+b+c)/c

=3+(b/a+a/b)+(b/c+c/b)+(c/a+a/c)

∵a/b+b/a≥2√(a/b*b/a)=2

（当a/b=b/a,a=b时,取等号）

同理：b/c+c/b≥2,a/c+c/a≥2

∴(b/a+a/b)+(b/c+c/b)+(c/a+a/c)≥6

∴3+(b/a+a/b)+(b/c+c/b)+(c/a+a/c)≥9

即1/a+1/b+1/c≥9