Opštinsko takmičenje (Municipal round) 2013 · Lemma

1GeometryProofIn a triangle \(ABC\), the altitude from \(A\) meets the line \(BC\) at \(D\), and its length satisfies \[ AD = \tfrac{1}{2}\,BC. \] Prove that the angle of the triangle at the vertex \(A\) cannot be obtuse. …2AlgebraFree answerFind the remainder on dividing the polynomial \(x^{2011} + 1\) by the polynomial \((x+1)^2\).3Number theoryProofDoes there exist a natural number \(n\) for which the decimal expansion of \(n!\) has the form \[ n! = \ldots 2012\,\underbrace{00\ldots 0}_{k}, \] that is, ends in the digit block \(2012\) followed by …4GeometryProofA convex quadrilateral \(ABCD\) satisfies \[ \angle DAB + \angle ABC = 120^\circ. \] Points \(P\) and \(Q\) are chosen so that the triangles \(ACP\) and \(BDQ\) are equilateral, with \(P\) lying in the …5CombinatoricsFree answerWhat is the largest number of chips that can be placed on the cells of a \(7 \times 7\) board so that no rectangle of area \(6\), with sides running along the grid lines, contains more than one chip?

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