IMO Shortlist 1975 problem 5
Dodao/la:
arhiva2. travnja 2012. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the set of all positive integers that do not contain the digit
![9](/media/m/7/f/0/7f02ff2403dbf63ddc4395762441de88.png)
(base
![10](/media/m/5/b/e/5beb46430dbe2d22c0f8289c36a92c84.png)
). If
![x_1, \ldots , x_n](/media/m/1/5/2/1522c8622b293f06ca4fafb1a3fd3bcd.png)
are arbitrary but distinct elements in
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
, prove that
%V0
Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that
$$\sum_{j=1}^n \frac{1}{x_j} < 80 .$$
Izvor: Međunarodna matematička olimpijada, shortlist 1975