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Putnam 2003 A4

  1. Oct 26, 2007 #1
  2. jcsd
  3. Oct 26, 2007 #2

    When they say B = 0 that has nothing to do with a generality argument. That is something you can say. Because if you complete the square on RHS you have just Nx^2+D with no middle term. That is what they mean by "shifting x" so if you shift x by x - B/A then you end up with a quadradic missing a middle term.

    The only generality argument is that a>0 or a<0. So say that a>0. In that case it must mean that A>=a>0 because otherwise LHS is a quadradic of a bigger coefficient.
     
  4. Oct 26, 2007 #3
    I see for the first part. But you mean shift by x - B/2A, right?

    But I do not see why you keep generality when you assume a > 0. Why do you not have to consider the case where a<0?
     
  5. Oct 26, 2007 #4
    Because the proof for a>0 and a<0 are almost identical. For definiteness it is easier to solve one of these cases. The other remaining case is similar.
    http://en.wikipedia.org/wiki/Without_loss_of_generality
     
  6. Oct 26, 2007 #5
    On second thought, I do not see why it is readily apparant that we can set B = 0. When you change coordinates to x - B/2A, how do you know that you are still in case 2? Maybe that is true, but it seems like that they would need to prove that...
     
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