Opštinsko takmičenje (Municipal round) 2010 · Lemma

1GeometryProofLet \(T_1\) and \(T_2\) be the centroids of triangles \(A_1B_1C_1\) and \(A_2B_2C_2\), respectively. Prove that \[ \overrightarrow{A_1A_2} + \overrightarrow{B_1B_2} + \overrightarrow{C_1C_2} = 3\,\overrightarrow{T_1T_2}. \] …2Number theoryFree answerA natural number \(n\) leaves the remainder \(35\) on division by \(2009\), and also leaves the remainder \(35\) on division by \(2010\). What remainder does \(n\) leave on division by \(42\)?3GeometryProofLet \(CD\) be the bisector of the angle \(BCA\) of a triangle \(ABC\), where \(D\) lies on the side \(AB\), and suppose that \[ AC + BD = BC + AD. \] Prove that the triangle \(ABC\) is isosceles.4Number theoryProofDoes the number \[ 2010^{2010} + 10^{2011} \] have more digits in its decimal representation than the number \(2010^{2010}\)?5CombinatoricsFree answerEach unit cell of a \(3 \times 3\) table is coloured with one of three colours. How many such colourings are there in which every two cells sharing a side are coloured differently?

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