Compete · Serbia
Opštinsko takmičenje (Municipal round) 2019
1Let \(AA_0\), \(BB_0\) and \(CC_0\) be the altitudes of a triangle \(ABC\), and let \(H\) be its orthocenter. Let \(M\) be the midpoint of the segment \(AA_0\) and let \(N\) be the midpoint of the segment …2Find all pairs of real numbers \(a\) and \(b\) such that the equality \[ \lfloor ax + by \rfloor + \lfloor bx + ay \rfloor = (a+b)\lfloor x+y \rfloor \] holds for all real numbers \(x\) and \(y\). (For …3Prove that in the regular octagon \(A_1A_2A_3A_4A_5A_6A_7A_8\) the diagonals \(A_1A_6\), \(A_3A_7\) and \(A_5A_8\) pass through one point.4Find all natural numbers \(n\) for which the three numbers \[ n-4, \qquad 2n+2, \qquad 4n+1 \] are all perfect cubes.5In the game Minesweeper, mines are placed on some cells of an \(a \times b\) board (\(a, b \in \mathbb{N}\)), and on every remaining cell one writes the number of neighbouring cells that contain a mine. …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.