very nice problem.. please, give me a solution

Valentin Vornicu wrote: Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that \[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } n. \] I just wonder whether Valentin did it knowingly. I have analyzed some cases and encountered some problems with it. Indeed, first of all I invented solution for all $n$ satisfying the Pell's equation $x^2-5(n+1)^2=-4$ . After that, the great idea come to me and I improved my construction, so I think it covers all sufficiently large $n$ (maybe $n>20$ or something like that). It is really "a relatively hard problem". The right stetement is the following (thanks to Bogdan Enescu for RMC2004 book ) Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that \[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } 2. \] Do you see the difference?

Please, post your complete solution Very nice problem =D

For which one? The idea I used for the initial problem (with $\frac{\dots}{n}$) is the following. We put $a_i=i$ all $i\leq n/2$ and $a_i=i$ or $a_i=2i$ for $i>n/2$.

i don´t understand your solution

It wasn't a complete solution, but just an idea. Note, that LHS depends only on the count of whose $a_i$, which are equal to $2i$. Using that we should adjust RHS. (and again, it is an idea )

If you complete your solution, please, post

I have it for all sufficietly large $n$. I am too lazy to write it...

If you find one little solution, please, post

Does anyone have a complete solution of the original problem?

Myth wrote: Valentin Vornicu wrote: Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that \[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } n. \] I just wonder whether Valentin did it knowingly. I have analyzed some cases and encountered some problems with it. Indeed, first of all I invented solution for all $n$ satisfying the Pell's equation $x^2-5(n+1)^2=-4$ . After that, the great idea come to me and I improved my construction, so I think it covers all sufficiently large $n$ (maybe $n>20$ or something like that). It is really "a relatively hard problem". The right stetement is the following (thanks to Bogdan Enescu for RMC2004 book ) Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that \[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } 2. \] Do you see the difference? My bad ... I did it this time Anyway it's great you solved the problem I accidentaly created

How do you do it. Bomb