Compete · Slovenia
Drzavno tekmovanje (national round) 2013
1Find all triples of prime numbers \(p\), \(q\), \(r\) for which \[ p + q^{2} = r^{4} . \]2For a real number \(a\), let \([a]\) denote the largest integer that is not greater than \(a\). Find all integers \(y\) for which there exists a real number \(x\) satisfying \[ \left[\frac{x+23}{8}\right] = \left[\sqrt{x}\,\right] = y . \] …3A triangle \(ABC\) has a point \(D\) on side \(AB\) and a point \(E\) on side \(AC\) such that \[ |AE| = |ED| = |DB| \qquad\text{and}\qquad |AD| = |DC| = |CB| . \] Determine the angles of triangle \(ABC\). …4We want to cover a \(4 \times 4\) board with tiles of the shape shown below, rotations and reflections being allowed. The tiles are permitted to overlap one another and to stick out beyond the edge of …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.