Republicko takmicenje (Republic round) 2005 · Lemma

1CombinatoricsFree answerHow many isosceles trapezoids with integer side lengths have perimeter \(2005\)? (A trapezoid here means a quadrilateral with exactly two parallel sides, so a parallelogram is not one. Trapezoids with …2GeometryProofLet \(ABC\) be an isosceles triangle with \(AB = AC\). Let \(D\) be the point of the side \(AC\) for which \(CD = 2 \, AD\), and let \(P\) be a point of the segment \(BD\) with \(\angle APC = 90^\circ\). …3GeometryProofLet \(A_1, A_2, \dots, A_{501}\) be arbitrary pairwise distinct points of the plane. Prove that on every circle of radius \(4\) there is a point \(M\) for which \[ MA_1 + MA_2 + \dots + MA_{501} \geq 2004 . \] …4AlgebraProofProve that for all real numbers \(a\) and \(b\), \[ a(1 + b^2) + b(1 + a^2) \leq (1 + a^2)(1 + b^2) . \]5CombinatoricsProofThe number \(1\) is written on a board \(2005\) times. A move consists of erasing two of the numbers written on the board and writing, in their place, one quarter of their sum. The move is repeated until …

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