Opštinsko takmičenje (Municipal round) 2023 · Lemma

1Logic and setsProofConsider the set of five Serbian words (written in the Latin alphabet) \[ X = \{\ \text{aca},\ \text{konac},\ \text{lopte},\ \text{loto},\ \text{prst}\ \}, \] and define two relations on \(X\): for words …2AlgebraFree answerA polynomial \(P\) with integer coefficients satisfies \[ P\bigl(P(2023) + 2023\bigr) = 1. \] Which values can the number \(P(2023)\) take?3GeometryProofLet \(ABC\) be a triangle. The tangents to the circumcircle of \(ABC\) at the points \(B\) and \(C\) intersect at a point \(X\). The circle through \(A\), \(B\), \(X\) meets the line \(BC\) again at a …4Number theoryFree answerDetermine all natural numbers \(n\) for which the number \[ n^2 + 7n + 2 \] is equal to a product of several (at least two) consecutive natural numbers.5CombinatoricsProofFor natural numbers \(m\) and \(n\), consider a board of dimensions \(m \times n\) made up of \(mn\) unit squares. Call the skeleton of the board the set of all unit segments that are edges of at least …

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