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Mr. Bee moderators
25/06/2018 at 02:53
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A polygon is a closed figure whose sides are straight lines. The above figure shows a six-sided polygon (a hexagon). Show that the sum of the angles of a hexagon, \(S^o\), is \(S^o= 540^o\).

Then, show that the generalized formula for the sum of the angles of an n-sided polygon is \(S^o= 180^o(n-2)\).

PolygonIGCSE

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    Alone 25/06/2018 at 07:10

    A B C D E F

    \(S^o=\)\(\angle\)BAF + \(\angle\)ABC + \(\angle\)BCD + \(\angle\)CDE + \(\angle\)DEF + \(\angle\)EFA

      =\(\angle\)AFB + \(\angle\)BAF + \(\angle\)ABF + \(\angle\)CBF + \(\angle\)BCF + \(\angle\)DCF + \(\angle\)CDF + \(\angle\)EDF + \(\angle\)DEF + \(\angle\)EFD + \(\angle\)CFD + \(\angle\)BFC

     =\(180^o+180^o+180^o+180^o=720^o\)

    Selected by MathYouLike
  • ...
    Mr. Bee moderators 25/06/2018 at 07:31

    Sory, I mistyped. It should've been \(720^o\) not \(540^o.\)


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Lê Quốc Trần Anh Coordinator
22/06/2018 at 07:36
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A bird collection has exactly four types of birds (eagles, doves, crows and sparrows). The eagles and doves make up 60% of the collection, and the doves and crows make up 20% of the collection. If the 18 crows in the collection represent 5% of the total number of birds, how many of the birds are sparrows? 


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Lê Quốc Trần Anh Coordinator
22/06/2018 at 07:36
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A 20-foot-high rectangular room has a floor that measures 18’ by 15’. Its doorway measures 3’ by 12’, and its only window measures 7’ by 10’. How many square feet of wall space does the room have?  


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Lê Quốc Trần Anh Coordinator
22/06/2018 at 07:36
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A square and a circle overlap such that a vertex of the square is at the center of the circle. The 4-inch radius of the circle is one-half the length of a side of the square. What is the area of the portion of the square region that is outside the circular region? Express your answer in terms of π.

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    huynh anh phuong 02/08/2019 at 14:01

    OMG !I don't khow what do you say!

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    A.R.M.Y 09/08/2018 at 15:55

    I don't know I'm sorry but i'm grade 4 =)


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Lê Quốc Trần Anh Coordinator
11/06/2018 at 02:11
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Prove that the product of three consecutive integers is not a square number.


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Lê Quốc Trần Anh Coordinator
13/06/2018 at 02:05
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Given \(a^3+b^3=2\) Prove that \(a+b\le2\)

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    Nguyễn Mạnh Hùng 13/06/2018 at 09:35

    We have:

    \(a^3+b^3=\left(a+b\right)\left(a^2+ab+b^2\right)=2\) (*)

    Other way:

    \(a^2+ab+b^2=a^2+2.\dfrac{1}{2}b+\left(\dfrac{1}{2}b\right)^2+\dfrac{3}{4}b^2\)

    \(=\left(a+\dfrac{1}{2}b\right)^2+\dfrac{3}{4}b^2\)

    Because \(\left(a+\dfrac{1}{2}b\right)^2\ge0\) with \(\forall a,b\)

    \(\dfrac{3}{4}b^2\ge0\) with \(\forall b\)

    So \(\left(a+\dfrac{1}{2}b\right)^2+\dfrac{3}{4}b^2\ge0\) with \(\forall a,b\)

    or \(a^2+ab+b^2\ge0\) with \(\forall a,b\) (**)

    From (*) and (**) we have  \(a+b\le2\)

    Your ex is so hard to do :) 

    Lê Quốc Trần Anh selected this answer.
  • ...
    Lê Thành 23/06/2018 at 05:27

    We have:

    a3+b3=(a+b)(a2+ab+b2)=2a3+b3=(a+b)(a2+ab+b2)=2 (*)

    Other way:

    a2+ab+b2=a2+2.12b+(12b)2+34b2a2+ab+b2=a2+2.12b+(12b)2+34b2

    =(a+12b)2+34b2=(a+12b)2+34b2

    Because (a+12b)2≥0(a+12b)2≥0 with ∀a,b∀a,b

    34b2≥034b2≥0 with ∀b∀b

    So (a+12b)2+34b2≥0(a+12b)2+34b2≥0 with ∀a,b∀a,b

    or a2+ab+b2≥0a2+ab+b2≥0 with ∀a,b∀a,b (**)

    From (*) and (**) we have  a+b≤2a+b≤2

    Your ex is so hard to do :) 


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Lê Quốc Trần Anh Coordinator
13/06/2018 at 02:07
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Prove that with a,b,c > 0: \(\dfrac{a^2}{b^2}+\dfrac{b^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{a}\)


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Lê Quốc Trần Anh Coordinator
09/06/2018 at 03:41
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In June, Casey counted the months until he would turn 16, the minimum age at which he could obtain his driver’s license. If the number of months Casey counted until his birthday was 45, in what month would Casey turn 16? 


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Lê Quốc Trần Anh Coordinator
09/06/2018 at 04:36
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Given \(P\left(x\right)+3P\left(2\right)=5x^2\left(\forall x\right)\). Calculate P(3)

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    Alone 13/06/2018 at 10:16

    With x=2 then \(4P\left(2\right)=5.2^2=20\Rightarrow P\left(2\right)=5\)

    With x=3 then \(P\left(3\right)+3P\left(2\right)=5.3^2=45\Rightarrow P\left(3\right)=45-5.3=30\)

    Lê Quốc Trần Anh selected this answer.

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Lê Quốc Trần Anh Coordinator
10/06/2018 at 06:31
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Find x,y,z satisfys: \(\dfrac{x}{8}=\dfrac{y}{3}=\dfrac{z}{10}\) and xy + yz + zx = 1206

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    Alone 14/06/2018 at 01:07

    \(\dfrac{x}{8}=\dfrac{y}{3}=\dfrac{z}{10}\Rightarrow\dfrac{x^2}{64}=\dfrac{xy}{24}=\dfrac{yz}{30}=\dfrac{zx}{80}=\dfrac{xy+yz+zx}{24+30+80}=\dfrac{1206}{134}=9\)

    \(\Rightarrow x^2=64.9=8^2.3^2\Rightarrow x=\pm24\)

    With x=24 then y=24:8.3=9;z=30

    With x=-24 then y=-9;z=-30

    Lê Quốc Trần Anh selected this answer.
  • ...
    Nguyễn Mạnh Hùng 15/06/2018 at 01:35

    We have:

    \(\dfrac{x}{8}=\dfrac{y}{3}=\dfrac{z}{10}\)

    \(\Rightarrow\dfrac{x^2}{64}=\dfrac{xy}{24}=\dfrac{yz}{30}=\dfrac{xz}{80}\)

    Apply the same sequence properties, we have:

    \(\dfrac{x^2}{64}=\dfrac{xy}{24}=\dfrac{yz}{30}=\dfrac{xz}{80}=\dfrac{xy+yz+xz}{24+30+80}=\dfrac{1206}{134}=9\)

    So \(x^2=9.64=576\)\(\Rightarrow x=24\)

    \(\dfrac{xy}{24}=9\Rightarrow xy=24.9=216\Rightarrow y=9\)

    \(\dfrac{yz}{30}=9\Rightarrow yz=270\Rightarrow z=30\)

    So \(\left(x;y;z\right)=\left(24;9;30\right)\)


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