Compete · Serbia
Opštinsko takmičenje (Municipal round) 1996
1Let \(p\) be a number such that \(p\) and \(p^{2} + 2\) are both prime. Prove that \(p^{3} + 2\) is prime as well.2Find all pairs of integers \(p\) and \(q\) for which the inequalities \[ p^{2} + q^{2} < 18p - 20q - 166, \qquad 32p - q^{2} > p^{2} + 12q + 271 \] hold simultaneously.3Let \(A\) be a subset of the set \(\{1, 4, 7, \dots, 1996\}\) containing exactly \(335\) elements. Prove that \(A\) contains two distinct numbers whose sum equals \(2000\).4Find all real numbers \(x\) for which \[ \bigl|\,|x| - 1\,\bigr| + \bigl|\,|x| + 2\,\bigr| = 3. \]5Two players alternately take balls from two boxes. On each turn, a player chooses one of the boxes and removes any number of balls from it (at least one). The player who takes the last ball wins. The first …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.