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March 27, 2017

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if abc=1 prove that 1/(1+a+b^-1) + 1/(1+b+c^-1) +1/(1+c+a^-1) =1

  • maths (exponents - ,

    Well, just using brute force, we have

    b/(b+ab+1) + c/(c+bc+1) + a/(a+ac+1)

    expand all that over a common denominator, and you get

    (a^2b^2c + a^bc^2 + 2a^2bc + a^2b + ab^2c^2 + 2ab^2c + 2abc^2 + 6abc + 2ab + ac^2 + 2ac + a + b^2c + 2bc + b + c)
    /
    (a^2b^2c^2 + a^2b^2c + a^2bc^2 + 2a^2bc + a^2b + ab^2c^2 + 2ab^2c + 2abc^2 + 4abc + 2ab + ac^2 + 2ac + a + b^2c + 2bc + b + c + 1)

    now replace all the abc occurrences with just 1, and you have

    (ab + ac + 2a + a^2b + bc + 2b + 2c + 6 + 2ab + ac^2 + 2ac + a + b^2c + 2bc + b + c)
    /
    (ab + ac + 2a + a^2b + bc + 2b + 2c + 6 + 2ab + ac^2 + 2ac + a + b^2c + 2bc + b + c)

    where the top and bottom are clearly the same.

    Now, for a trick of evaluation which avoids all that mess, I haven't come up with one yet.

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