Izbirno tekmovanje (selection round) 2004 · Lemma

1Number theoryFree answerFind all natural numbers \(m\) and \(n\) that satisfy \[ \frac{3}{m} + \frac{5}{n} = 1 . \]2GeometryFree answerIn the isosceles right triangle \(ABC\) the right angle is at \(C\) and each leg has length \(2\). A circular arc \(\ell\) centred at \(A\) divides the triangle into two parts of equal area, and a circular …3GeometryFree answerA rectangle \(ABCD\) has \(|AB| = 2a\) and \(|AD| = a\). Let \(E\) be the midpoint of the side \(AB\), and choose an arbitrary point \(F\) on the side \(AD\). The area of the triangle \(ECF\) depends on …4AlgebraFree answerFind all solutions of the equation \[ x = \bigl|\,2x - |60 - 2x|\,\bigr| . \]5Number theoryFree answerNika and Tim played a series of card games. A draw was impossible. They agreed in advance that after each game the winner receives more points than the loser and that the loser receives a positive number …

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