Drzavno tekmovanje (national round) 2018 · Lemma

1GeometryMultiple choiceFive brothers - Jure, Klemen, Luka, Miha and Nace - bought a bar of chocolate. When they unwrapped it, they found that it had snapped into the seven pieces shown below, so they shared those seven pieces …2AlgebraMultiple choiceA real number \(a\) satisfies \(a^{2} - \tfrac{1}{2}a = \tfrac{1}{4}\). What is the value of \(a^{3} - \tfrac{1}{2}a\)? A \(-\tfrac{1}{4}\) B \(\tfrac{1}{4}\) C \(\tfrac{1}{2}\) D \(4\) E \(\tfrac{1}{8}\) …3GeometryMultiple choiceThe horizontal segment in the picture is divided into six parts of equal length, and every triangle appearing in the picture is equilateral. The whole figure is shaded in two colours, light grey and dark …4Number theoryFree answerFind all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]5GeometryFree answerThree squares are inscribed in a right triangle \(ABC\) with the right angle at \(B\), arranged as in the picture: the largest one has two sides lying on the legs of the triangle, and each of the two smaller …6CombinatoricsFree answerSixteen points of the integer lattice are marked, as in the picture: all points \((x,y)\) with \(x\) and \(y\) taken from \(\{1,2,3,4\}\). At most how many of these points can be coloured red so that no …

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