Drzavno takmicenje (National round) 2007 · Lemma

1GeometryFree answerLet \(ABCDEF\) be a regular hexagon. A point \(M\) is taken on the diagonal \(AC\) and a point \(N\) on the diagonal \(CE\) so that both are placed at the same relative position: \[ \frac{AM}{AC} = \frac{CN}{CE} = \lambda . \] …2GeometryProofThree points \(A\), \(B\), \(C\), not lying on one line, are given. Construct a point \(D\) for which the quadrilateral \(ABCD\) is at the same time cyclic and tangential, that is, admits both a circumscribed …3Number theoryFree answerFor a natural number \(n\), let \(x_n\) be the number obtained by writing the natural numbers from \(1\) to \(n\) one after another, for example \[ x_{15} = 123456789101112131415 . \] Find all natural …4AlgebraFree answerFind the smallest possible value of \(x + y + z\) for non-negative real numbers \(x, y, z\) subject to \[ \begin{aligned} xy(10x + 10y + 7z) &\geq 27, \\ yz(10y + 10z + 7x) &\geq 27, \\ zx(10z + 10x + 7y) &\geq 27. \end{aligned} \] …5CombinatoricsFree answerIn the plane of a triangle \(ABC\) one draws \(n\) lines, each of them parallel to one of the three sides of the triangle. Determine the smallest \(n\) for which these \(n\) lines can cut the plane into …

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