Compete · Serbia
Drzavno takmicenje (National round) 2009
1How many functions \(f \colon \mathbb{R}^{+} \to \mathbb{R}\) are there such that \[ f\left(x + \frac{1}{x}\right) = x^{2} + \frac{1}{x^{2}} \] holds for every \(x \in \mathbb{R}^{+}\)?2Let \(C\) be a point of the segment \(AB\) other than \(A\) and \(B\), and let \(k_{0}, k_{01}, k_{02}\) be the circles with diameters \(AB, AC\) and \(CB\) respectively. Let \(D\) be a point in which …3Finitely many arcs are marked on a circle. The length of each of them is smaller than half of the circumference, and any three of the marked arcs have a common point. Prove that there is a point of the …4The figure below consists of a triangle \(ABC\) together with the circular segment erected on the side \(BC\), on the opposite side of \(BC\) from \(A\). Construct at least one straight line that divides …5Does there exist a natural number which is a perfect square and whose sum of digits equals \(2008^{2009}\)?
Work through the paper in order - each problem opens in the workspace with this paper as its trail.