Drzavno tekmovanje (national round) 2023 · Lemma

1AlgebraMultiple choiceWhat is the value of \[ 2^{0^{2^{3}}} + 0^{2^{3^{2}}} + 2^{3^{2^{0}}} + 3^{2^{0^{2}}} \, ? \] A \(3\) B \(4\) C \(7\) D \(12\) E Greater than \(100\).2GeometryMultiple choiceIn triangle \(ABC\) the angle at \(A\) is a right angle. Points \(D\), \(E\) and \(F\) are chosen on the sides \(AB\), \(BC\) and \(CA\) respectively, so that \[ |BD| = |BE| \qquad\text{and}\qquad |CF| = |CE| , \] …3AlgebraMultiple choiceA goldsmith owns two alloys. The first is \(90\%\) gold, the second \(54\%\) gold. He melts together \(320\) g of the first alloy and \(160\) g of the second to obtain a new alloy. What percentage of gold …4Number theoryFree answerDetermine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.5GeometryFree answerA disc \(K\) of radius \(R\) is cut into three circular sectors in such a way that the areas of the two smaller sectors add up to the area of the largest sector, while the difference of the areas of the …6CombinatoricsFree answerLevel 1 of the computer game Zakladnica takes place in an underground treasury built from \(13\) octagonal and \(12\) square rooms, arranged as in the figure. The only way into the treasury, and the only …

Work through the paper in order - each problem opens in the workspace with this paper as its trail.