# 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 $s_1<s_2<...<s_n$ 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.

• Note that the RHS is not in lowest terms: it's equal to $\frac{17}{670}$. – Erick Wong Jun 3 '17 at 7:33

## 2 Answers

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.