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Quoc Tran Anh Le Coordinator
28/02/2019 at 15:13
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What is the average of the prime numbers between 20 and 30?

 

Mathcounts

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    Huy Toàn 8A (TL) 01/03/2019 at 03:54

    We have :  primes from 20 to 30 are 23,29

    => total : 23 + 29 = 52

    The answer í 52

    Quoc Tran Anh Le selected this answer.

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Quoc Tran Anh Le Coordinator
28/02/2019 at 14:44
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What is the volume of a rectangular prism of height 5 cm, width 7 cm and depth 3 cm? 

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    Nguyễn Viết Trung Nhân 17/03/2020 at 02:33

    The volume of the rectangular prism is:

          5 x 7 x 3 = 105(cm2)


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Blue Moon
28/02/2019 at 14:24
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Cho tứ giác ABCD nội tiếp (O). Gọi E là giao điểm của AB và CD, F là giao điểm của AC và BD. Đường tròn ngoại tiếp tam giác BDE cắt đường tròn ngoại tiếp tam FDC tại điểm K khác D. Tiếp tuyến của (O) Tại C và B cắt nhau tại M.

1. Chứng minh tứ giác BKCM nội tiếp.

2. Chứng minh E, M, F thẳng hàng.

moon

  • ...
    Quoc Tran Anh Le Coordinator 28/02/2019 at 14:41

    In this page, Vietnamese is not allowed, post an English question and we'll help.


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Uchiha Sasuke
28/02/2019 at 12:27
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The sum of all solutions of the equation \(\left|2x+|x-3|\right|=2\) is:...

sum


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Quoc Tran Anh Le Coordinator
27/02/2019 at 11:14
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If two more than three times x is equal to five less than ten times x, what is the value of x ?

 

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Quoc Tran Anh Le Coordinator
27/02/2019 at 11:14
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Ted flips a coin that is equally likely to land heads up or tails up. Ted flips the coin 10 times, and each time it lands heads up. What is the probability that the next flip will also land heads up? Express your answer as a common fraction.

 

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Quoc Tran Anh Le Coordinator
27/02/2019 at 11:13
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Bob has 40 cents in his pocket. If Bob has no pennies, how many different combinations of quarters, dimes and/or nickels could he have?

 

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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 09:55
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30) Let a, b, c are the length of a triangle. Prove that:

\(\dfrac{ab}{a+b-c}+\dfrac{bc}{b+c-a}+\dfrac{ca}{c+a-b}\ge a+b+c\)

inequality

  • ...
    FacuFeri 28/02/2019 at 15:58

    Put \(A=\dfrac{ab}{a+b-c}+\dfrac{bc}{b+c-a}+\dfrac{ac}{c+a-b}\)

    Because a ; b ; c are the length of a triangle so \(a+b-c;b+c-a;c+a-b>0\)

    Put \(a+b-c=x;b+c-a=y;c+a-b=z\)

    \(\Rightarrow\dfrac{x+y}{2}=b;\dfrac{y+z}{2}=c;\dfrac{x+z}{2}=a;a+b+c=x+y+z\) 

    We have : \(\dfrac{\left(x+y\right)\left(x+z\right)}{2.2x}+\dfrac{\left(x+y\right)\left(y+z\right)}{2.2y}+\dfrac{\left(x+z\right)\left(y+z\right)}{2.2z}\)

    \(=\dfrac{x\left(x+y+z\right)+yz}{4x}+\dfrac{y\left(x+y+z\right)+xz}{4y}+\dfrac{z\left(x+y+z\right)+xy}{4z}\)

    \(=\dfrac{x+y+z}{4}+\dfrac{x+y+z}{4}+\dfrac{x+y+z}{4}+\dfrac{yz}{4x}+\dfrac{xz}{4y}+\dfrac{xy}{4z}\)

    \(=\dfrac{3\left(x+y+z\right)}{4}+\dfrac{y^2z^2}{4xyz}+\dfrac{x^2z^2}{4xyz}+\dfrac{x^2y^2}{4xyz}\)

    Applying the inequality \(a^2+b^2+c^2\ge ab+bc+ac\) , we have :

    \(A\ge\dfrac{3\left(x+y+z\right)}{4}+\dfrac{xyz\left(x+y+z\right)}{4xyz}=x+y+z=a+b+c\)

    Equal sign occurs \(\Leftrightarrow x=y=z\Leftrightarrow a=b=c\)

     
    Lê Anh Duy selected this answer.
  • ...
    Nguyễn Thị Linh 06/03/2019 at 23:34

    FacuFeri copy internet


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Lê Anh Duy
27/02/2019 at 10:19
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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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