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Math Help - Inequality problem

  1. #1
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    Smile Inequality problem

    If a,b > 0 and a+b=1
    prove that

    (a+\frac{1}{a})^2+(b+\frac{1}{b})^2 \ge \frac{25}{2}

    Where I can find more problems like that ?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by gilyos View Post
    If a,b > 0 and a+b=1
    prove that

    (a+\frac{1}{a})^2+(b+\frac{1}{b})^2 \ge \frac{25}{2}

    Where I can find more problems like that ?
    The function f(x)=\left(x+\tfrac{1}{x}\right)^2 is convex. That pretty much finishes it. Look up IMO.
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  3. #3
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    Imo ?!
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by gilyos View Post
    Imo ?!
    International Math Olympiad.
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  5. #5
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    in the IMO the problems are difficult ... i need something like this problem
    and if u can explain who I prove this problem
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by gilyos View Post
    in the IMO the problems are difficult ... i need something like this problem
    and if u can explain who I prove this problem
    Oh...I'm sorry that you don't want hard problems. Do you know what convexity is? Is this for a class? I mean, we need a little more.
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  7. #7
    Super Member PaulRS's Avatar
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    Quote Originally Posted by gilyos View Post
    and if u can explain who I prove this problem
    Do you know Jensen's Inequality? I think that is what Drexel28 had in mind.
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  8. #8
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    Where I can find inequalites problems with Cauchy , Bernoulli's inequality , Inequality of arithmetic and geometric means ??? Book or site or something else ...
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  9. #9
    Super Member PaulRS's Avatar
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    See this book for example, it's excellent.
    Here you have Olympiad Inequalities by Thomas Mildford, and here : Topics in Inequalities by Hojoo Lee.
    There are several pages with inequality problems, just google a bit.

    Have fun
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