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Thread: GLaw’s challenge problems

  1. #16
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    Re: GLaw’s challenge problems

    Spoiler:
    $a,\ b,\ c,\ x,\ y\ are\ all\ real\ and \ge 0.$

    $STEP\ I:$

    $0 \le \sqrt{xy} \le \dfrac{x + y}{2} \implies 0 \le \dfrac{x + y}{2} - \sqrt{xy} \implies$

    $0 \le \left(\dfrac{x + y}{2} - \sqrt{xy}\right)^2 = \left(\dfrac{x + y}{2}\right)^2 - 2\sqrt{xy}\left(\dfrac{x + y}{2}\right) + xy \implies$

    $0 \le \left(\dfrac{x + y}{2}\right)^2 - (x + y)\sqrt{xy}+ xy \implies$

    $(x + y)\sqrt{xy} \le \left(\dfrac{x + y}{2}\right)^2 + xy.$

    $STEP\ II:$

    $\dfrac{a^2 + b^2 + c^2 + 3(ab + bc + ac)}{2} = \dfrac{2a^2 + 2b^2 + 2c^2 + 6(ab + bc + ac)}{4} =$

    $\dfrac{(a^2 + 2ab + b^2) + (a^2 + 2ac + c^2) + (b^2 + 2bc + c^2) + 4(ab + bc + ac)}{4} =$

    $\left\{\left(\dfrac{a + b}{2}\right)^2 + ab\right\} + \left\{\left(\dfrac{a + c}{2}\right)^2 + ac\right\} +\left\{\left(\dfrac{b + c}{2}\right)^2 + bc\right\}.$

    $STEP\ III:$

    $By\ STEP\ I:$

    $(a + b)\sqrt{ab} \le \left(\dfrac{a + b}{2}\right)^2 + ab,\ and$

    $(a + c)\sqrt{ac} \le \left(\dfrac{a + c}{2}\right)^2 + ac,\ and$

    $(b + c)\sqrt{bc} \le \left(\dfrac{b + c}{2}\right)^2 + bc.$

    $STEP\ IV:$

    $Combining\ STEPS\ II\ and\ III:$

    $ (a + b)\sqrt{ab} + (a + c)\sqrt{ac} + (b + c)\sqrt{bc} \le \dfrac{a^2 + b^2 + c^2 + 3(ab + bc + ac)}{2}.$

    I had the general idea right away, but finding the specific route took a while. Thanks for the challenge.

    I still think you should put a new challenge into a new thread.
    Thanks from GLaw
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  2. #17
    Member GLaw's Avatar
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    Re: GLaw’s challenge problems

    Thanks, Jeff.

    Quote Originally Posted by JeffM View Post
    I still think you should put a new challenge into a new thread.
    I’m out of challenge problems for now but if I have a new one I’ll start a new thread.

    Solution to Problem #5:
    Spoiler:
    Expand $\left(\sqrt a-\sqrt b\right)^4+\left(\sqrt b-\sqrt c\right)^4+\left(\sqrt c-\sqrt a\right)^4\geq0$.
    Last edited by GLaw; Feb 7th 2016 at 02:54 PM.
    Thanks from topsquark and JeffM
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  3. #18
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    Re: GLaw’s challenge problems

    You can borrow this one:
    why is number 18 a good pick for me right now?
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  4. #19
    Forum Admin topsquark's Avatar
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    Re: GLaw’s challenge problems

    Quote Originally Posted by DenisB View Post
    You can borrow this one:
    why is number 18 a good pick for me right now?
    Oh! Is your girlfriend finally legal now?

    -Dan
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  5. #20
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    Re: GLaw’s challenge problems

    Quote Originally Posted by topsquark View Post
    Oh! Is your girlfriend finally legal now?
    Dan me good man, when you made that post,
    19 would have been a good pick for you: why?
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  6. #21
    Forum Admin topsquark's Avatar
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    Re: GLaw’s challenge problems

    Quote Originally Posted by DenisB View Post
    Dan me good man, when you made that post,
    19 would have been a good pick for you: why?
    Oh, I don't know, I can think of about 20 reasons.

    -Dan
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  7. #22
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    Re: GLaw’s challenge problems

    Because it was post#19 !!
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