IMO Shortlist 1995 problem A2
Dodao/la:
arhiva2. travnja 2012. Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
be non-negative integers such that
![ab \geq c^2,](/media/m/a/d/6/ad679d6cf85af300f4ce96e54eef4efe.png)
where
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
is an integer. Prove that there is a number
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
and integers
![x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n](/media/m/b/2/d/b2dc6c88c1238be70753245bbabaf70f.png)
such that
%V0
Let $a$ and $b$ be non-negative integers such that $ab \geq c^2,$ where $c$ is an integer. Prove that there is a number $n$ and integers $x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that
$$\sum^n_{i=1} x^2_i = a, \sum^n_{i=1} y^2_i = b, \text{ and } \sum^n_{i=1} x_iy_i = c.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1995