There are precisely two nonnegative real values of R, so that for each of these values:

³√(3 + √R) + ³√(3 - √R) is an integer.

Find these values, and prove that no other nonnegative real R can conform to the given conditions.

*** While the solution may be facile with the aid of a computer program, show how to derive it without one.

using the equations given on this page I found the following solutions  R=242/27 and R=368/27.  I used the fact that for r=0 the equation starts at a value between 2 and 3 and as r goes to infinity it tends to zero.  thus the only integer values it could take for postive r would be either 1 or 2.  I used the equation given on the linked page to solve for the appropriate values that would give the values of 1 and 2.

Posted by Daniel on 2008-08-08 07:03:55