Okruzno takmicenje (Regional round) 2004 · Lemma

1GeometryFree answerThe lengths of the sides of a triangle \(ABC\) are three consecutive natural numbers. The median drawn from \(A\) is perpendicular to the bisector of the angle \(\angle ABC\). Determine the lengths of …2GeometryProofLet \(H\) be the orthocentre of an acute triangle \(ABC\), and let \(A_1\), \(B_1\) and \(C_1\) be the centres of the circles circumscribed about the triangles \(BHC\), \(CHA\) and \(AHB\) respectively. …3AlgebraFree answerConsider the finite sequence of \(2003\) numbers given by \[ a_n = \left\lfloor \frac{n^2}{2004} \right\rfloor, \qquad n = 1, 2, \ldots, 2003, \] where \(\lfloor x \rfloor\) denotes the greatest integer …4Number theoryProofDoes there exist a polynomial \(P\) with integer coefficients for which \[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \] …5CombinatoricsFree answerA merchant has to ferry seven goods across a river: a piece of cheese, a mouse, a rat, a cat, a dog, a wolf and a bear. His boat has room for only \(k\) of the seven at a time. If they are left without …

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