Izbirno tekmovanje (selection round) 2008 · Lemma

1AlgebraFree answerFind every real number \(x\) that satisfies \[ \Bigl|\, \bigl|\, |2-x| - x \,\bigr| - 8 \,\Bigr| \le 2008 . \]2GeometryProofLet \(ABCD\) be a quadrilateral and let \(K\) be a point inside triangle \(ABD\) such that triangles \(ABD\) and \(KCD\) are similar, the vertices corresponding in the written order. Prove that triangles …3Number theoryFree answerFind the smallest three-digit number with the property that every digit of its triple is even.4GeometryFree answerThe equilateral triangle \(ABC\) has sides of length \(4\) cm. Let \(D\) be the midpoint of the side \(AB\), and let \(E\) and \(F\) be the feet of the perpendiculars dropped from \(D\) to the sides \(BC\) …5CombinatoricsProofAt a national competition the students worked on \(4\) problems. Each problem was marked with a whole number of points, at least \(0\) and at most \(7\). Altogether \(42\) students competed. Exactly half …

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