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Math Help - Proving

  1. #1
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    Proving

    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.
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  2. #2
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    Re: Proving

    Assume a,b,c\not = 1, otherwise the problem is ill defined.

    Let x=\frac{1}{a}, y=\frac{1}{b} , z=\frac{1}{c}.

    Then your original condition becomes x+y+z=3, and the expression you want to equate to zero becomes

    \frac{xyz}{(1-y)(1-z)}+\frac{xyz}{(1-x)(1-z)}+\frac{xyz}{(1-x)(1-y)}

    =xyz\left(\frac{(1-x)+(1-y)+(1-z)}{(1-x)(1-y)(1-z)}\right)

    =xyz\left(\frac{3-(x+y+z)}{(1-x)(1-y)(1-z)}\right)

    Substituting x+y+z=3 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.
    Thanks from agentmulder
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