Compete · Serbia
Opštinsko takmičenje (Municipal round) 2020
1Three distinct points \(A\), \(B\), \(C\) lie on a line \(\ell\), and a point \(O\) lies off \(\ell\). The perpendicular bisectors of the segments \(OA\), \(OB\) and \(OC\) form a triangle \(EFG\). Prove …2A snake starts in the upper-left cell of a \(2 \times n\) board, where \(n\) is a natural number. From one cell it may move to another whenever the two cells share an edge, but it may never visit a cell …3Find all three-element sets \(A\) with the following two properties: (i) the set \(A\) has at least two elements in common with its power set \(\mathcal{P}(A)\); (ii) \(3 \in A\). (It is understood that …4Let \(m > 1\) be a natural number. Prove that there is no sequence of \(2^{m}\) consecutive natural numbers all of which have exactly \(m\) prime factors, counted with multiplicity. (For example, the number …5Is it possible to divide a square into convex pentagons?
Work through the paper in order - each problem opens in the workspace with this paper as its trail.