Opštinsko takmičenje (Municipal round) 2006 · Lemma

1CombinatoricsFree answerIn how many ways can three rooks be placed on a chessboard of dimensions \(6 \times 2006\) so that no two of them attack each other? (Two rooks attack each other when they stand in the same row or in the …2Number theoryFree answerDetermine the greatest common divisor of the numbers \(2^{2006}-1\) and \(2^{2004}-1\).3Number theoryFree answerThe sum of \(49\) natural numbers equals \(999\). Find the largest possible value of their greatest common divisor.4GeometryFree answerIn an isosceles triangle, the bisector of one of the angles at the base is exactly twice as long as the altitude drawn to that base. Determine the angles of the triangle.5GeometryProofCan an equilateral triangle be divided - that is, actually cut up with scissors - into \(2006\) equilateral triangles?

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