Compete · Serbia
Opštinsko takmičenje (Municipal round) 2012
1Let \(ABCD\) be a square and let \(E\) be the midpoint of its side \(CD\). The line through \(D\) perpendicular to the diagonal \(BD\) meets the line \(AE\) at a point \(F\). Prove that the points \(B\), …2Tomorrow's weather forecast consists of the following three claims: it will be cloudy, or it will snow, or the wind will blow; if it is cloudy and snowing, then the wind will blow; if the wind does not …3Determine all natural numbers \(n\) for which the number of positive divisors of \(n^{3}\) is exactly \(2011\) greater than the number of positive divisors of \(n\).4In a triangle \(ABC\) the angle at \(B\) is obtuse, \(\angle ABC > 90^\circ\), and the side \(AC\) is twice as long as \(AB\), that is \(2\cdot AB = AC\). Prove that \[ 2\cdot\angle ACB > \angle BAC. \] …5A bishop on a chessboard attacks every square lying on one of the two diagonals through it. Call a square covered if a bishop stands on it or a bishop attacks it. Prove that seven bishops can never be …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.