Does there exist a rational $x_1$, such that all members of the sequence $x_1, x_2, \ldots, x_{2024}$ defined by $x_{n+1}=x_n+\sqrt{x_n^2-1}$ for $n=1, 2, \ldots, 2023$ are greater than $1$ and rational?

Let $ABCD$ be a convex quadrilateral with $\angle ABC=\angle ADC=120^{\circ}$. The point $E$ lies on the segment $AD$ and is such that $AE \cdot BC=AB \cdot DE$ and similarly the point $F$ lies on the segment $BC$ and satisfies $BF \cdot CD=AD \cdot CF$. Show that $BE$ and $DF$ are parallel.

Let $n \geq 2$ be a positive integer. There are $2n$ cities $M_1, M_2, \ldots, M_{2n}$ in the country of Mathlandia. Currently there roads only between $M_1$ and $M_2, M_3, \ldots, M_n$ and the king wants to build more roads so that it is possible to reach any city from every other city. The cost to build a road between $M_i$ and $M_j$ is $k_{i, j}>0$. Let $$K=\sum_{j=n+1}^{2n} k_{1,j}+\sum_{2 \leq i<j \leq 2n} k_{i, j}.$$Prove that the king can fulfill his plan at cost no more than $\frac{2K}{3n-1}$.

Let $n$ be a positive integer. A regular hexagon $ABCDEF$ with side length $n$ is partitioned into $6n^2$ equilateral triangles with side length $1$. The hexagon is covered by $3n^2$ rhombuses with internal angles $60^{\circ}$ and $120^{\circ}$ such that each rhombus covers exactly two triangles and every triangle is covered by exactly one rhombus. Show that the diagonal $AD$ divides in half exactly $n$ rhombuses.

The positive reals $a, b, c, x, y, z$ satisfy $$5a+4b+3c=5x+4y+3z.$$Show that $$\frac{a^5}{x^4}+\frac{b^4}{y^3}+\frac{c^3}{z^2} \geq x+y+z.$$ Proposed by Dominik Burek

Given is a prime number $p$. Prove that the number $$p \cdot (p^2 \cdot \frac{p^{p-1}-1}{p-1})!$$is divisible by $$\prod_{i=1}^{p}(p^i)!.$$