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Possibilities of a (Posted on 2012-12-07) Difficulty: 3 of 5
Suppose x, y, a are real numbers such that x+y = x^3 +y^3 = x^5 +y^5 = a. Find all possible values of a.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Analytical solution (spoiler) Comment 1 of 1
1) One solution is x = x^3 = x^5 and y = y^3 = y^5
    x can be 0, 1 or -1 and y can be 0, 1 or -1.
    so a can be -2, -1, 0 , 1 or 2

2) Are there any other solutions?  Well,
    (x+y) - (x^3 + y^3) = 0
    so x^3 - x = y^3 - y
    so x(x+1)(x-1) = y(y+1)(y-1)                 <= equation a

    also,  (x^3+y^3) - (x^5 + y^5) = 0
    so x^5 - x^3 = y^5 - y^3
    so (x^3)(x+1)(x-1) = (y^3)(y+1)(y-1)    <= equation b

    If x <> 0,1,-1 then equation a can be divided into equation b,
    so x^2 = y^2

    This yields another solution,  x = -y, which gives a = 0

    It also suggests x = y as a solution, but substituting into
    (x+y) - (x^3 + y^3) = 0  gives x = 0 or 1 or -1, which is not a new solution.

SO, FINAL ANSWER, a can only be -2, -1, 0 , 1 or 2

  Posted by Steve Herman on 2012-12-07 16:46:03
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