My Math Forum A problem for the year 2012

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 March 6th, 2012, 07:21 PM #1 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 A problem for the year 2012 Well, I have been looking through some old problems from famous math contests and a lot of them seem to have problems that have the number of the current year in them, so in the spirit of these math contests I came up with such a problem. I actually was unable to solve the last challenge that I posted. I kept approaching it in different ways and thinking that I had a proof, but there was always something wrong with it. However, in my efforts to solve that problem I discovered a few patterns, and this problem is the result of my hard work: Prove that the equation $\sum_{k=1}^{501} x_k^{251} = 2012y$ has no solutions in positive integers $x_k, y$ such that none of the $x_k$ are divisible by 503.

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