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Math Help - Cyclically 130

  1. #1
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    Cyclically 130

    Consider all quadruples of real numbers (x1,x2,x3,x4), such that x1x2x3x4 and each of the four numbers plus the product of the other three equals 130. If S is the sum of all possible values of x4, what is S,
    rounded to the nearest integer
    ?
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  2. #2
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    Re: Cyclically 130

    1,2,3,4 are the subscripts of x
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  3. #3
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    Re: Cyclically 130

    Hello, geniusgarvil!

    Consider all quadruples of real numbers a,b,c,d , such that a \le b \le c \le d

    and each of the four numbers plus the product of the other three equals 130.

    If S is the sum of all possible values of d, what is S? .(Round to nearest integer.)

    We have: . \begin{Bmatrix}a + bcd \:=\:130 \\ b + acd \:=\:130 \\ c + abd \:=\:130 \\ d + abc \:=\:120 \end{Bmatrix}

    Since the four variables are completely interchangeable,
    . . we conclude that: . a \,=\, b \,=\, c \,=\, d.

    The first equation becomes: . a + a^3 \:=\:130 \quad\Rightarrow\quad a^3 + a - 130 \:=\:0
    . . The only real root is: . a \,=\,5.


    Therefore: . a \,=\,b\,=\,c\,=\,d\,=\,5 \quad\Rightarrow\quad S \,=\,5
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  4. #4
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    Re: Cyclically 130

    I,don't know what's wrong in your solution, but the answer's incorrect.
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