Compete · Serbia
Opštinsko takmičenje (Municipal round) 2024
1It is known that the number \[ 21982145917308330487013369 \] is equal to \(n^{13}\) for some natural number \(n\). Determine \(n\).2Aca and Branko play the following game on a \(2023 \times 2024\) board. First Aca chooses a square of the board and places a queen on it. Then the players move the queen alternately, following the rules …3Let \[ N = 1^{n} + 2^{n} + 3^{n} + 4^{n}, \qquad n \in \mathbb{N}. \] What is the greatest number of zeros in which the number \(N\) can end?4Let \(X\) be the midpoint of the base \(AB\) of a trapezoid \(ABCD\) (\(AB \parallel CD\)). Prove that if \[ \angle ADX = \angle BCX, \] then the bisectors of the angles \(\angle ADX\), \(\angle DXC\) …5The infinite sequence of natural numbers \(a_1, a_2, a_3, \dots\) is defined by \[ a_1 = a_2 = 1, \qquad a_{n+2} = a_{n+1} + a_n \ \text{ for every } n \in \mathbb{N}. \] Prove that \[ \frac{a_1}{2} + \frac{a_2}{2^2} + \dots + \frac{a_{2024}}{2^{2024}} < 2. \] …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.