if 1/a+1/b+1/c=3 then prove that 1/a(b-1)(c-1) + 1/b(a-1)(c-1) + 1/c(a-1)(b-1) is 0.
Assume , otherwise the problem is ill defined.
Let .
Then your original condition becomes , and the expression you want to equate to zero becomes
Substituting makes this whole expression zero.
You may have noticed I skipped a few steps here and there, but if you are doing these kind of questions you should be competent enough to work out the details.