Compete · Slovenia
Izbirno tekmovanje (selection round) 2009
1A competition paper consisted of \(24\) multiple-choice questions. A contestant who circled no answer to a question, or circled more than one, received \(0\) points for it; a circled correct answer was …2In triangle \(ABC\) the sides satisfy \(|AB| = 2|AC|\). A point \(D\) is chosen on side \(AB\) and a point \(E\) on side \(BC\) so that \(\angle BAE = \angle ACD\). The segments \(AE\) and \(CD\) intersect …3Prove that the inequality \[ x^2 + y^2 + 1 \ge 2\bigl(xy - x + y\bigr) \] holds for every pair of real numbers \(x\) and \(y\). When does equality occur?4Find all rational numbers \(r\) and all integers \(k\) that satisfy \[ r\bigl(5k - 7r\bigr) = 3 . \]5Every school of a certain region sent exactly \(3\) students to a competition, and Andrej, Blaz and Zan all came from the same school. When the competitors lined up to collect their starting numbers, Andrej …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.