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Math Help - Prove positive real numbers a,b,c.

  1. #16
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    Re: Prove positive real numbers a,b,c.

    Well it would equal 3xyz/(a-b)^3 + (b-c)^3 + (c-a)^3 wouldnt it? I dont know how to simplify the denomiator.
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  2. #17
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    Re: Prove positive real numbers a,b,c.

    Quote Originally Posted by pikachu26134 View Post
    Well it would equal 3xyz/(a-b)^3 + (b-c)^3 + (c-a)^3 wouldnt it? I dont know how to simplify the denomiator.
    see post #9 for numerator and post#12 for denominator. then post the answer again. don't use x,y and z in the answer. use only a, b and c.
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  3. #18
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    Re: Prove positive real numbers a,b,c.

    [3(a^2-b^2)(b^2-c^2)(c^2-a^2)]/ [3(a-b)(b-c)(c-a)] is that right?
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  4. #19
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    Re: Prove positive real numbers a,b,c.

    Quote Originally Posted by pikachu26134 View Post
    [3(a^2-b^2)(b^2-c^2)(c^2-a^2)]/ [3(a-b)(b-c)(c-a)] is that right?
    perfect. now can you further simplify this. use (a^2-b^2)=(a-b)(a+b). etc. what do you get?
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  5. #20
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    Re: Prove positive real numbers a,b,c.

    You then get (a+b)(b+c)(c+a) > 8abc
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  6. #21
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    Re: Prove positive real numbers a,b,c.

    Quote Originally Posted by pikachu26134 View Post
    You then get (a+b)(b+c)(c+a) > 8abc
    can you please post that here in more detail.
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  7. #22
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    Re: Prove positive real numbers a,b,c.

    [(a+b)(a-b)(b+c)(b-c)(c-a)(c+a)]/[(a-b)(b-c)(c-a)] and then you simply it to [(a+b)(b+c)(c+a)]/1 = (a+b)(b+c)(c+a)
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  8. #23
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    Re: Prove positive real numbers a,b,c.

    Quote Originally Posted by pikachu26134 View Post
    [(a+b)(a-b)(b+c)(b-c)(c-a)(c+a)]/[(a-b)(b-c)(c-a)] and then you simply it to [(a+b)(b+c)(c+a)]/1 = (a+b)(b+c)(c+a)
    that's perfect!
    now can i have some rep points.
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  9. #24
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    Re: Prove positive real numbers a,b,c.

    Yup, ive thanked on all your posts. What do I do from there? I presume its the quadratic rule where either (a+b)>8abc etc.... but what exactly do i do to find sets abc?
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  10. #25
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    Re: Prove positive real numbers a,b,c.

    Quote Originally Posted by pikachu26134 View Post
    Yup, ive thanked on all your posts. What do I do from there? I presume its the quadratic rule where either (a+b)>8abc etc.... but what exactly do i do to find sets abc?
    (\sqrt{a}- \sqrt{b})^2>0
    \Rightarrow a+b-2 \sqrt{ab}>0

    \Rightarrow a+b>2 \sqrt{ab} .......(1)

    similarly b+c> 2 \sqrt{bc} ......(2)

    and c+a > 2 \sqrt{ca} ...... (3)
    multiply the above three inequalities.
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  11. #26
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    Re: Prove positive real numbers a,b,c.

    2abc+a^2b+ac^2+a^2c+b^2c+ab^2+bc^2 > 2sqr ab x 2sqr bc x 2sqr ca
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  12. #27
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    Re: Prove positive real numbers a,b,c.

    Quote Originally Posted by pikachu26134 View Post
    2abc+a^2b+ac^2+a^2c+b^2c+ab^2+bc^2 > 2sqr ab x 2sqr bc x 2sqr ca
    well what i was expecting was
    (a+b)(b+c)(c+a)> 2 \sqrt{ab} \cdot 2 \sqrt{bc} \cdot 2 \sqrt{ca}=8 \sqrt{abbcca}=8abc
    so (a+b)(b+c)(c+a)>8abc
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  13. #28
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    Re: Prove positive real numbers a,b,c.

    Oh right sorry, I thought you meant expanding the whole lot. Yes, that is correct. But now what do I do? I realise you simplified the RHS to get 8abc.
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  14. #29
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    Re: Prove positive real numbers a,b,c.

    Quote Originally Posted by pikachu26134 View Post
    Oh right sorry, I thought you meant expanding the whole lot. Yes, that is correct. But now what do I do? I realise you simplified the RHS to get 8abc.
    what is left? which part do you think remains unproven?
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  15. #30
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    Re: Prove positive real numbers a,b,c.

    It remains to find groups abc such that that equation is true
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