IMO Shortlist 1992 problem 1
Dodao/la:
arhiva2. travnja 2012. Prove that for any positive integer
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
there exist an infinite number of pairs of integers
![(x, y)](/media/m/1/5/2/1520b43353795b60686f7df83802e90a.png)
such that
(i)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
are relatively prime;
(ii)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
divides
![x^2 + m](/media/m/8/c/c/8ccec04dcbc8c0d2ad221254cf93dc60.png)
;
(iii)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
divides
(iv)
![x + y \leq m + 1-](/media/m/a/2/f/a2f18b2c2f22bf12c20c2495add97c0d.png)
(optional condition)
%V0
Prove that for any positive integer $m$ there exist an infinite number of pairs of integers $(x, y)$ such that
(i) $x$ and $y$ are relatively prime;
(ii) $y$ divides $x^2 + m$;
(iii) $x$ divides $y^2 + m.$
(iv) $x + y \leq m + 1-$ (optional condition)
Izvor: Međunarodna matematička olimpijada, shortlist 1992