IMO Shortlist 1982 problem 19
Dodao/la:
arhiva2. travnja 2012. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the set of real numbers of the form
![\frac{m+n}{\sqrt{m^2+n^2}}](/media/m/c/1/2/c120997c324f0e7196bf0c00d8a7cdac.png)
, where
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
are positive integers. Prove that for every pair
![x \in M, y \in M](/media/m/1/e/0/1e05813ac7184afdaa57aa10ea1ee31e.png)
with
![x < y](/media/m/0/8/b/08b3f86331faf64ac3ba1d2d58aacb0a.png)
, there exists an element
![z \in M](/media/m/4/5/b/45b61f1c0e8116c2fed898368e44a291.png)
such that
%V0
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$
Izvor: Međunarodna matematička olimpijada, shortlist 1982