Compete · Serbia
Opštinsko takmičenje (Municipal round) 2021
1Eight players took part in a chess tournament, and each of them played exactly one game against every other participant. A win earns \(1\) point, a loss \(0\) points, and a draw \(0.5\) points for each …2Let \(A\) and \(B\) be non-empty sets, neither of which is a subset of the other. For a natural number \(n\) consider the equality \[ \underbrace{A \setminus \bigl(B \setminus (A \setminus (B \setminus \cdots))\bigr)}_{n \text{ sets}} \;=\; \underbrace{A \mathbin{\triangle} \bigl(B \mathbin{\triangle} (A \mathbin{\triangle} (B \mathbin{\triangle} \cdots))\bigr)}_{n \text{ sets}} \] …3Let \(xOy\) be an angle, and let \(A\), \(B\) and \(C\) be points on the arm \(Ox\) such that \(OA = 3\), \(OB = 4\) and \(OC = 6\). Let \(D\) be the foot of the perpendicular dropped from \(B\) to the …4Let \(n \ge 3\), and suppose \(n\) consecutive odd three-digit numbers are given. Prove that these \(n\) numbers can be arranged into a sequence \(b_1, b_2, \ldots, b_n\) so that the number \[ \overline{b_1b_2\ldots b_n}, \] …5The cells of an \(n \times n\) table are to be coloured with \(n\) different colours in such a way that every row and every column contains cells of all \(n\) colours. Determine the smallest and the largest …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.