Izbirno tekmovanje (selection round) 2005 · Lemma

1Number theoryFree answerFind the smallest natural number \(n\) for which the sum \[ n + 2n + 3n + \cdots + 9n \] is a number whose decimal representation has all of its digits equal.2AlgebraFree answerFor which values of the parameter \(a\) does the system of equations \[ |x - 1| + |y - a| = 1 , \qquad y = -2|x - 1| - 1 \] have exactly three solutions \((x, y)\)?3AlgebraFree answerThe real numbers \(a\) and \(b\) satisfy \[ a^3 = 3ab^2 + 11 , \qquad b^3 = 3a^2 b + 2 . \] Compute the value of \(a^2 + b^2\).4GeometryFree answerThe rhombus \(ABCD\) has an acute interior angle at the vertex \(A\). The perpendicular dropped from \(D\) to the side \(AB\) meets it at the point \(E\), which splits the side into the two pieces \[ |AE| = x , \qquad |EB| = y . \] …5GeometryFree answerAn equilateral triangle \(ACG\), a regular pentagon \(CDEFG\) and a regular octagon \(BHIJKLDC\) all meet at the point \(C\), as shown in the figure. Determine the angles of the triangle \(ABC\).

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