Opštinsko takmičenje (Municipal round) 2026 · Lemma

1AlgebraFree answerLet \(a\), \(b\), \(c\), \(d\) and \(e\) be integers such that \[ a \ne b, \qquad b \ne c, \qquad c \ne d, \qquad d \ne e, \qquad e \ne a. \] Determine the smallest possible value of the expression \[ I = a^{2} + b^{2} + c^{2} + d^{2} + e^{2}. \] …2Number theoryProofLet \(a_{1}, a_{2}, \ldots, a_{n}\) be pairwise distinct numbers from the set \(\{1, 2, \ldots, n\}\), where \(n \in \mathbb{N}\). Prove that the number \[ (a_{1} - 1) + (a_{2} - 2)^{2} + \cdots + (a_{n} - n)^{n} \] …3GeometryProofTwo circles \(k_{1}\) and \(k_{2}\) intersect at two distinct points, and \(AB\) is their common chord. A point \(P\) is chosen on \(k_{1}\) so that it lies outside \(k_{2}\). The lines \(PA\) and \(PB\) …4Number theoryFree answerFind all natural numbers that are powers of \(3\) and whose representation in base \(12\) contains only the digits \(6\) and \(9\).5CombinatoricsFree answerThe numbers \(1, 2, 3, 4, 5, 6, 7, 8\) are split into three disjoint nonempty sets. Let \(P_{1}\), \(P_{2}\) and \(P_{3}\) be the products of the numbers in the first, the second and the third set, respectively, …

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