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Math Help - Inequality with the side lengths of a triangle

  1. #1
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    Inequality with the side lengths of a triangle

    a, b, c - the side lengths of a triangle
    k - a positive real number

    \frac{a^{k}}{(b+c)^{2}}+\frac{b^{k}}{(a+c)^{2}}+\f  rac{c^{k}}{(b+a)^{2}}\geq \frac{1}{2}(\frac{a^{k}}{a^{2}+bc}+\frac{b^{k}}{b^  {2}+ac}+\frac{c^{k}}{c^{2}+ba})

    I observed that: (b+c)^{2}+(a+c)^{2}+(b+a)^{2}=2(a^{2}+bc+b^{2}+ac+  c^{2}+ba), so CBS inequality seems a good idea, but what I tried to do didn't help me ._.

    I want some indications. Thanks in advance.
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  2. #2
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    I observed:

    a =< b =< c < a+b (since a triangle)

    Your equation:
    Equality obtained if a = b = c , else greater than
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  3. #3
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    I used Cebīșev's inequality (supposing that a<=b<=c) and I got this:

    \sum_{a, b, c}\frac{1}{(a+b)^2}\geq \frac{1}{2}\sum_{a, b, c}\frac{1}{a^2+bc}

    And no, even if it looks easy, I don't know how to prove it >.>
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