Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation \[n^3+m^3-nm(n+m)=2023.\]

In a triangle, the edges are extended past both vertices by the length of the edge opposite to the respective vertex. Show that the area of the resulting hexagon is at least $13$ times the area of the original triangle.

For a competition a school wants to nominate a team of $k$ students, where $k$ is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the $n \ge 2$ students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition: $(*)$ If a student $X$ is not nominated for the team, there is a student $Y$ on the team who defeated $X$ in at least two disciplines. Determine all positive integers $n \ge 2$ such that for any combination of rankings, a team can be chosen to satisfy the condition $(*)$, when a) $k=2$, b) $k=3$.

Determine all triples $(a,b,c)$ of real numbers with \[a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.\]

Let $ABC$ be an acute triangle with altitudes $AA'$ and $BB'$ and orthocenter $H$. Let $C_0$ be the midpoint of the segment $AB$. Let $g$ be the line symmetric to the line $CC_0$ with respect to the angular bisector of $\angle ACB$. Let $h$ be the line symmetric to the line $HC_0$ with respect to the angular bisector of $\angle AHB$. Show that the lines $g$ and $h$ intersect on the line $A'B'$.

The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.