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Minimum

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Lê Anh Duy
27/02/2019 at 10:07
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31) Let x,y are 2 positive real integers such that \(\sqrt{xy}\cdot\left(x-y\right)=x+y\)

Find the minimum value of \(P\) = x + y

Minimum

  • ...
    FacuFeri 10/05/2019 at 11:48

    hiu We have : \(\sqrt{xy\left(x-y\right)}=x+y\Leftrightarrow xy\left(x-y\right)^2=\left(x+y\right)^2\)

    \(xy\left(x-y\right)^2=\dfrac{1}{4}.4xy\left[\left(x+y\right)^2-4xy\right]\le\dfrac{\left(x+y\right)^4}{16}\)

    so \(\left(x+y\right)^4\ge16\left(x+y\right)^2\)  \(\Leftrightarrow p^4-16p^2\ge0\Leftrightarrow p\ge4\)

    Equal sign occurs \(\Leftrightarrow x=2+\sqrt{2};b=2-\sqrt{2}\)


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Lê Anh Duy
27/02/2019 at 10:19
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5
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32) Let x,y > 0 such that x + y \(\ge\) 10.

Find the minimum value of \(P=2x+y+\dfrac{30}{x}+\dfrac{5}{y}\)

Minimum

  • ...
    Huy Toàn 8A (TL) 01/03/2019 at 04:38

    \(P=\left(\dfrac{30}{x}+\dfrac{6x}{5}\right)+\left(\dfrac{y}{5}+\dfrac{5}{y}\right)+\dfrac{4}{5}\left(x+y\right)\ge2.\sqrt{\dfrac{30}{x}+\dfrac{6x}{5}}+2.\sqrt{\dfrac{y}{5}+\dfrac{5}{y}}+\dfrac{4}{5}.10\)

    \(P=2.6+2.1+8=22\)

    Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}\dfrac{30}{x}=\dfrac{6x}{5}\\\dfrac{y}{5}=\dfrac{5}{y}\\x+y=10\end{matrix}\right.=>\left\{{}\begin{matrix}x^2=25\\y^2=25\\x+y=10\end{matrix}\right.\)

    \(=>x=y=5\)

    \(Pmin=22< =>x=y=5\)

  • ...
    Huy Toàn 8A (TL) 28/02/2019 at 14:06

    \(P=2x+y+\dfrac{30}{x}+\dfrac{5}{y}=\left(x+y\right)\left(\dfrac{5}{x}+\dfrac{5}{y}\right)+\left(\dfrac{25}{x}+x\right)\)

    \(\left\{{}\begin{matrix}x>0=>x=\left(\sqrt{x}\right)^2\\y>0=>y=\left(\sqrt{y}\right)^2\end{matrix}\right.\)

    \(P1=x+y\ge10\)

    \(P2=\dfrac{5}{x}+\dfrac{5}{y}=5.\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\ge5.\dfrac{4}{x+y}=\dfrac{5.4}{10}=2\) khi \(x=y\)

    \(P3=\dfrac{25}{x}+x\ge2.\sqrt{\dfrac{25}{x}.x}=2.5=10\) khi \(x=5\)

    \(P=\sum P\ge10+2+10=22\) khi \(\left(x;y\right)=\left(5;5\right)\)

    Study well

  • ...
    Vui Ghét Nét 07/03/2019 at 03:53

    Reference Lê Anh Duy


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Huỳnh Anh Phương
06/11/2018 at 05:26
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Niushi has 13 packs of milk. She remember that: Her mom said her drinks milk every day and each day drinks 2 packs of milk in minimum, and 3 packs of milk in maximum. What is the minimum of days Niushi can drink with these pack of milk?

Minimum

  • ...
    Nguyen Thu An 31/01/2019 at 15:57

    5 days

    Pls vote for meyeu


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Lê Anh Duy
01/09/2018 at 16:02
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28) Given x2 + 5y2 - 4xy - x + 2y - 6 = 0

Find the max and min values of A = x - 2y

MaximumMinimummemorable equality

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    Alone 23/10/2018 at 13:01

    \(x^2+5y^2-4xy-x+2y-6=0\)

    \(\Leftrightarrow x^2-4xy+4y^2-\left(x-2y\right)-6=-y^2\)

    \(\Leftrightarrow\left(x-2y\right)^2-\left(x-2y\right)-6=-y^2\le0\)

    \(\Leftrightarrow\left(x-2y+2\right)\left(x-2y-3\right)\le0\)

    \(\Leftrightarrow-2\le x-2y\le3\)

    Lê Anh Duy selected this answer.

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Quoc Tran Anh Le Coordinator
05/08/2018 at 03:48
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Sam, Taylor and Pat counted the number of fish in each of their fish tanks. They noticed that Sam’s tank had exactly 25% more fish than Taylor’s tank, and Pat’s tank had exactly 24% more fish than Sam’s tank. If each tank had at least one fish, what is the minimum combined number of fish that could have been in the three tanks?

MathcountsMinimum


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Lê Anh Duy
31/07/2018 at 08:58
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23) There are two numbers: one has 6 divisors and one has 4 divisors. What is the minimum divisors of their product?

MinimumDivisor

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    Tôn Thất Khắc Trịnh 31/07/2018 at 14:11

    Mininum divisors? That is the number 1, obviously

    Or did you mean the minimum NUMBER of divisors?
    If so than it's 9, how you ask?
    To create less permutations with the divisors, the factors should overlap the most, hence, the 6 divisor number is a5 and the 4 divisor number is a3, so the product would be a8, having 9 divisors.


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Searching4You
26/07/2017 at 12:06
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Let a,b are numbers which satisfy \(\left\{{}\begin{matrix}a+b=3\\a^2+b^2\ge5\end{matrix}\right.\)

Find minimum value of the expression : \(P=a^4+b^4+6a^2b^2\)?

Minimum

  • ...
    Kaya Renger Coordinator 16/08/2017 at 21:35

    \(\left\{{}\begin{matrix}a+b=3\\a^2+b^2\ge5\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a^2+2ab+b^2=9\\a^4+2a^2b^2+b^4\ge25\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}2ab=9-\left(a^2+b^2\right)\\a^4+b^4+2a^2b^2\ge25\end{matrix}\right.\)

    We have : a2 + b2 \(\ge\) 5

    => \(-\left(a^2+b^2\right)\le5\)

    => \(9-\left(a^2+b^2\right)\ge9-5=4\)

    => \(2ab\ge4\)

    => \(ab\ge2\)

    <=> \(a^2b^2\ge4\)

    <=> \(4a^2b^2\ge16\)

    Plus \(4a^2b^2\ge16\) into \(a^4+b^4+2a^2b^2\ge25\)

    => \(a^4+b^4+6a^2b^2\ge41\)

    => Min = 41

    That is my opinion :v 

    Selected by MathYouLike
  • ...
    ¤« 03/04/2018 at 13:32

    {a+b=3a2+b2≥5⇒{a2+2ab+b2=9a4+2a2b2+b4≥25⇒{2ab=9−(a2+b2)a4+b4+2a2b2≥25

    We have : a2 + b2 ≥

     5

    => −(a2+b2)≤5

    => 9−(a2+b2)≥9−5=4

    => 2ab≥4

    => ab≥2

    <=> a2b2≥4

    <=> 4a2b2≥16

    Plus 4a2b2≥16

     into a4+b4+2a2b2≥25

    => a4+b4+6a2b2≥41

    => Min = 41

    That is my opinion :v

    There is something is wrong here 

    Change to :

    We have a2+b2≥5

    => 9−(a2+b2)≤4

    => 2ab≤4

    => ab≤2

    <=> a2b2≥4

    => 4a2b2≥16

  • ...
    Kaya Renger Coordinator 16/08/2017 at 22:06

    There is something is wrong here 

    Change to :

    We have \(a^2+b^2\ge5\)

    => \(9-\left(a^2+b^2\right)\le4\)

    => \(2ab\le4\)

    => \(ab\le2\)

    <=> \(a^2b^2\ge4\)

    => \(4a^2b^2\ge16\)


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Dung Trần Thùy
19/03/2017 at 22:06
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4
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Find the maximum and the minimum of \(N=\dfrac{4x-8}{x^2-4x+8}\).

MaximumMinimumFraction

  • ...
    ¤« 03/04/2018 at 13:33

    (*) 1+N=.....=x2/(x-2)2+4

    We have x2 ≥

     0 , (x-2)2+4 ≥

     4 > 0 

    So 1+N ≥

     0 => N ≥

     -1 ;equality : x=0 

    (*)1-N=....=(x-4)2/(x-2)2+4 

    ....... -> N ≤

     1 , equality : x=4

  • ...
    ¤« 03/04/2018 at 13:32

    There is something is wrong here 

    Change to :

    We have a2+b2≥5

    => 9−(a2+b2)≤4

    => 2ab≤4

    => ab≤2

    <=> a2b2≥4

    => 4a2b2≥16

  • ...
    Phan Văn Hiếu 28/03/2017 at 21:30

    trình bày = tiếng việt đc ko tiếng anh ngại viết lắm


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