Compete · Slovenia
Drzavno tekmovanje (national round) 2009
1Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \begin{aligned} x + y + 2z &= 0, \\ xy - z^2 &= 0, \\ y^2 + 5z + 6 &= 0. \end{aligned} \]2Find the smallest natural number \(n\) that is divisible by \(20\) and for which \(n^2\) is a perfect cube and \(n^3\) is a perfect square.3Let \(ABCD\) be a convex quadrilateral and let \(E\) and \(F\) be points on the sides \(AB\) and \(AD\) respectively, chosen so that \(EF \parallel BD\). The segment \(CE\) meets the diagonal \(BD\) at …4A natural number is written in every cell of a square table. Call the table interesting if the sum of all the numbers in it is odd and, in addition, the sum of the four numbers covered by any placement …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.