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Math Help - Here's a problem

  1. #1
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    Here's a problem

    I give you a bigrade: a^n + b^n + c^n = d^n + e^n + f^n; n = 1,2

    Now give me ab + ac + bc = de + df + ef (please show all work) and find values for a - f that makes the equation work.
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  2. #2
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    Quote Originally Posted by wonderboy1953 View Post
    I give you a bigrade: a^n + b^n + c^n = d^n + e^n + f^n; n = 1,2

    Now give me ab + ac + bc = de + df + ef (please show all work) and find values for a - f that makes the equation work.
    Do you know how to solve this? If not it does not belong in this forum

    CB
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  3. #3
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    Quote Originally Posted by wonderboy1953 View Post
    I give you a bigrade: a^n + b^n + c^n = d^n + e^n + f^n; n = 1,2

    Now give me ab + ac + bc = de + df + ef (please show all work) and find values for a - f that makes the equation work.

    we now have two equations

     a + b + c = e + d + f

     a^2 + b^2 + c^2 = e^2 + d^2 + f^2


    And do you know that

     \sum^n a_k^2 = \left ( \sum^n a_k \right )^2 - 2 \sum_{ i \neq j} a_i a_j


    a^2 + b^2 + c^2 = ( a + b + c )^2 - 2(ab+ bc + ca)

     d^2 + e^2 + f^2 = ( d + e + f )^2 - 2 (de + ef +fd )


    so we have  ( a + b + c )^2 - 2(ab+ bc + ca) =   ( d + e + f )^2 - 2 (de + ef +fd )


     ab + bc + ca = de + ef + fd
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  4. #4
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    Response to Captain Black

    Of course I know how to solve this problem, otherwise I wouldn't have posted it here (I didn't want to immediately post the solution as that would have given away the show).

    simplependulum got the answer right. I would have squared both sides and then use simple algebra to derive the end result to show that it follows naturally - the same difference, but I think my way would be more understandable. Now for the next part (which I know how to do):

    Starting from:

    , can you show through adding 1 to each and every member of this bigrade that you still have equality? [the first step is to derive (a + 1)^2 + (b + 1)^2 + (c + 1)^2 = (d + 1)^2 +          (e + 1)^2 + (f + 1)^2]
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  5. #5
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    BTW

    Here are one set of values (among the infinite) that make the bigrade work: a = 1, b = 6, c = 8, d = 2, e = 4 and f = 9. The reason why it's important to specify is you have can doubters (like Kronecker) who say that nothing is proven unless you can show everything.
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  6. #6
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    Solving the puzzle

    I left it to be proven that:



    reduces to:



    Opening parentheses, you get:

    a^2 + 2a + 1 + b^2 + 2b + 1 + c^2 + 2c + 1 = d^2 + 2d + 1 +<br />
e^2 + 2e + 1 + f^2 + 2f + 1

    Now rearranging terms and combining some:

    a^2 + b^2 + c^2 + 2(a + b + c) + 3 = d^2 + e^2 + f^2 + <br />
2(d + e + f) + 3

    Obviously the 3 cancels out. Also, by assumption, since this is a bigrade,
    a + b + c = d + e + f also cancels out from the equation (with the coefficient 2) which leaves you with:

    a^2 + b^2 + c^2 = d^2 + e^2 + f^2

    which is true by assumption. Here's my main point: with any multigrade, you can add any positive integer (to each and every term) and get a new multigrade, no exceptions.
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