What do you mean with the word <<square-free>> ?Thanks

I remember proving this for any $a>2$ instead of $2005$. It was a problem from an Iranian olympiad and TST and the topic was started by Omid Hatami. There should be enough good keywords here for anyone who wants to perform a search .

$n$ is called squarfree iff it isn't divisible by a perfect square $>1$, or equivalently every prime factor occurs at most twice in $n$.

I performed a search and here it is: http://www.mathlinks.ro/Forum/viewtopic.php?highlight=tst&t=30001

I believe all those similar problems were created upon the inspiration of IMO 2000\5

And that's why you solved it very successfully . (It's strange that only you and your brother were able to solve this problem. )

hy i wonder what were the results of the selection test? how many people solved 3 or more problems?