Izbirno tekmovanje (selection round) 2006 · Lemma

1AlgebraFree answerFind every real number \(x\) that satisfies the equation \[ \bigl|\,2006 - |206 - x|\,\bigr| = 26 . \]2Number theoryFree answerJana and Zana each wrote down her own age. Both ages turned out to be two-digit numbers written with the same two digits, only in the opposite order. Five years from now, Jana will be exactly twice as …3GeometryFree answerIn the right triangle \(ABC\) the right angle is at \(C\), and \(|AC| = 4\), \(|BC| = 8\). A point \(D\) is taken on the side \(BC\) so that \(|CD| = 5\). How far is \(D\) from the side \(AB\)?4GeometryProofThe incircle of a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(D\) and \(E\) respectively. Let \(F\) be any point of the side \(AB\) lying between \(A\) and \(D\), and let \(G\) …5AlgebraFree answerAndrej and Blaz drew a circle in the yard and marked two diametrically opposite points on it. Each of them stood on one of these points, and at the same instant they started walking around the circle in …

Work through the paper in order - each problem opens in the workspace with this paper as its trail.