Compete · Slovenia
Drzavno tekmovanje (national round) 2019
1The three presents in the picture are all boxes in the shape of a cuboid with edge lengths \(10\) cm, \(20\) cm and \(30\) cm. Reading from left to right, the ribbon tying them measures \(x\) cm, \(y\) …2Polona wants to draw three lines through one common point, as in the picture, so that the angles between them satisfy \(\beta = 2\alpha\) and \(\alpha = 3\gamma\). How many degrees must the angle \(\alpha\) …3What is the value of the expression \[ 2019^{3} - 3 \cdot 2019 \cdot 2018 - 2018^{3} \, ? \] A \(-1\) B \(0\) C \(1\) D \(2018\) E \(2019\)4Find all natural numbers \(n\) whose cube equals the sum of the squares of three divisors of \(n\), where the three divisors need not be different from one another.5Let \(ABC\) be a triangle. Points \(D\) and \(E\) lie on the rays \(CA\) and \(CB\) respectively, but not on the sides of the triangle \(ABC\), and are chosen so that \[ |AD| = |BE| = |AB| . \] Let \(G\) …6Find all pairs of real numbers \(x\) and \(y\) that satisfy the system \[ \frac{y^{3} + 15x^{2}}{y^{4} - x^{3}} = \frac{y^{2} + 15x}{y^{3} - x^{2}} , \] \[ \frac{1500y^{3} + 4x^{2}}{9y^{4} - 4} = \frac{1500y^{2} + 4x}{9y^{3} - 4} . \] …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.