Compete · Serbia
Okruzno takmicenje (Regional round) 2014
1Aca, Branka, Vera and Goran were each given the same kind of task by their mathematics teacher: divide one positive real number by another. Aca computed \(a_1 : a_2\), Branka computed \(b_1 : b_2\), Vera …2Two segments of lengths \(a\) and \(b\) are given. Construct a triangle \(ABC\) having these two segments as sides, in such a way that the angle opposite one of them is three times as large as the angle …3Determine all natural numbers \(k\), \(m\) and \(n\) for which \[ 2^k + 10^m - 10^n = 2014 . \]4In a triangle \(ABC\) the bisector of the angle at the vertex \(A\) meets the side \(BC\) at the point \(D\). The perpendicular dropped from \(B\) to the line \(AD\) meets the circumcircle of the triangle …5In the Mad Forest there lived \(6\) werewolves, \(17\) unicorns and \(55\) spiders. A werewolf can eat a spider or a unicorn, but not another werewolf; a spider can eat a unicorn, but neither a werewolf …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.