Compete · Slovenia
Regijsko tekmovanje (regional round) 2012
1Peter keeps horses and cows on his farm. To begin with he had exactly as many horses as cows, and that common number was greater than \(0\). He then bought some more cows, so that the number of cows rose …2At most how many interior angles of a polygon with \(n\) vertices can be greater than \(180^\circ\)? (The polygon is simple: its sides meet only at the shared endpoints of neighbouring sides.) A \(n - 1\) …3The number whose cube equals \(2012^{12}\) was multiplied by the square of the number \(2012^{11}\). Which number was obtained? A \(2012^{58}\) B \(2012^{26}\) C \(2012^{88}\) D \(2012^{15}\) E \(2012^{12}\) …4In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at \(D\), and the bisector of the angle \(\angle CBA\) meets the side \(AC\) at \(E\). Suppose that \(|CD| = |CE|\). …5Find all natural numbers \(n\) and all primes \(p\) for which \[ \sqrt{\,n + \frac{p}{n}\,} \] is a natural number.6Lara and Sara draw \(n\) straight lines on a rectangular sheet of paper, taking turns and drawing one line each time. Every line is parallel to one of the edges of the sheet and runs from edge to edge, …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.