Opštinsko takmičenje (Municipal round) 2002 · Lemma

1Number theoryProofLet \(n\) be a natural number. Prove that \(3n^2 + 3n + 7\) is not the cube of any natural number.2GeometryProofIn triangle \(ABC\) the angle at \(B\) equals \(60^\circ\). The bisector of \(\angle CAB\) meets the opposite side at \(D\), the bisector of \(\angle BCA\) meets the opposite side at \(E\), and \(S\) is …3AlgebraFree answerDetermine all functions \(f : \mathbb{R} \to \mathbb{R}\) such that for every real number \(x\), \[ f(x+1) \le x \le f(x) + 1. \]4CombinatoricsFree answerLet \(S = \{1, 2, \ldots, 20\}\). What is the largest possible number of elements of a subset \(A \subseteq S\) with the property that \(2x \notin A\) whenever \(x \in A\)?5CombinatoricsFree answerIn a handball tournament every team played exactly one match against each of the other teams. A win is worth \(2\) points, a loss \(0\), and a drawn match gives \(1\) point to each of the two teams. The …

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