IMO Shortlist 2010 : N1 - Finding the sequence Find the least positive integer $n$ for which there exists a set
  $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive
  integers such that $$ \left( 1 - \frac{1}{s_1} \right) \left( 1 -
 \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) =
 \frac{51}{2010}.$$


Here is my approach, WLOG $s_1&lt;s_2&lt;...&lt;s_n$ and $s_i\ge i+1$, thus $\prod (1-\frac{1}{s_i})\ge \prod\frac{i}{i+1}=\frac{1}{n+1}$ thus $n\ge 39$. 
But I can't find a sequence of length 39 to finish off. 

Please show me how did you think of a sequence. There is a similar problem , but my question is basically "what's the motivation for finding such a sequence?". So show your scratch work(Not Literally) or thought process if possible.
Take the 39 numbers $2,3,4,\cdots,40$ and replace 34 with 67:
$$\frac{1-\frac{1}{67}}{1-\frac{1}{34}}\prod_{k=2}^{40}\left(1-\frac{1}{k}\right)=
\frac{2\cdot 34}{67}\cdot \frac{1}{40}=\frac{17}{670}= \frac{51}{2010}$$
There's an entire thread on AoPS about this problem.