Compete · Serbia
Opštinsko takmičenje (Municipal round) 2005
1In a triangle \(ABC\), let \(C_1\) be the midpoint of the side \(AB\), so that \(CC_1\) is the median from \(C\). Let \(K\) be the midpoint of the segment \(CC_1\), and let the line \(AK\) meet the side …2Find all prime numbers \(p\), \(q\), \(r\), not necessarily different from one another, and all positive integers \(n\), for which \[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{n}. \]3Determine all triples of integers \((x, y, z)\) that satisfy \[ x^{2} + y^{2} + z^{2} = 2004\,xyz . \]4Let \(S = \{s, i, c, g\}\). a) How many relations on \(S\) are not symmetric? b) How many antisymmetric relations are there on \(S\)?5Prove or disprove the following assertion. Among any six positive integers it is always possible to choose three of them that are pairwise coprime, or three of them that have a common divisor greater than …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.