Compete · Serbia
Drzavno takmicenje (National round) 2011
1For each \(n = 1, 2, 3, \ldots\) Perica looks for the smallest block of \(2n+1\) consecutive positive integers with the property that the sum of the squares of the smallest \(n+1\) of them equals the sum …2Let \(H\) and \(O\) be the orthocentre and the circumcentre of a triangle \(ABC\) with \(AB \neq AC\). The lines \(AH\) and \(AO\) meet the circumcircle of \(ABC\) a second time at \(M\) and \(N\) respectively. …3Call a positive integer symmetric if its decimal representation reads the same from left to right as from right to left. Prove that there are infinitely many positive integers \(n\) for which the numbers …4For which positive integers \(m\) and \(n\) can an \(m \times n\) rectangle be covered completely and without overlaps by copies of the three figures shown below, each of them built from unit squares? …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.