APMO

For the sake of simplicity, let us set $k=2009$.

First of all, choose $n$ distinct positive integers $b_1,\cdots ,b_n$ suitably so that their product is a pure $k+1$-th power (for example, let $b_i=i^{k+1}$ for $i=1,\cdots ,n$). Then we have $b_1\cdots b_n=t^{k+1}$ for some positive integer $t$. Set $b_1+\cdots +b_n=s$.

Now we set $a_i=b_i{s}^{k^2-1}$ for $i=1,\cdots ,n$, and show that $a_1,\cdots ,a_n$ satisfy the required conditions. Since $b_1,\cdots ,b_n$ are distinct positive integers, it is clear that so are $a_1,\cdots ,a_n$. From $$\begin{array}{rl}{a}_1+\cdots +a_n& =s^{k^2-1}\left(b_1+\cdots +b_n\right)=s^{k^2}={\left(s^k\right)}^{2009}\\ a_1\cdots a_n& ={\left(s^{k^2-1}\right)}^n{b}_1\cdots b_n={\left(s^{k^2-1}\right)}^n{t}^{k+1}={\left(s^{(k-1)n}t\right)}^{2010}\end{array}$$ we can see that $a_1,\cdots ,a_n$ satisfy the conditions on the sum and the product as well. This ends the proof of the assertion.