Compete · Slovenia
Drzavno tekmovanje (national round) 2005
1The real numbers \(x\), \(y\), \(z\) satisfy \(xyz = 1\). Compute the value of \[ \frac{x+1}{xy+x+1} + \frac{y+1}{yz+y+1} + \frac{z+1}{zx+z+1}. \]2Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.3Let \(I\) be the incenter of a triangle \(ABC\), and suppose that \[ |CA| + |AI| = |BC| . \] Determine the ratio of the sizes of the angles \(\angle BAC\) and \(\angle CBA\).4Ales, Brane and Cvetka made a large pile of cards, writing on each card one of the numbers \(2, 3, 4, 5, 6, 7, 8\); every one of these numbers appears on many cards. Maja, who arrived later, picked three …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.