Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$$$\sqrt{y^3 - z} = x - 1$$$$\sqrt{z^3 - x} = y - 1$$

In a trapezoid $ABCD$ whose parallel sides $AB$ and $CD$ are in ratio $\frac{AB}{CD}=\frac32$, the points $ N$ and $M$ are marked on the sides $BC$ and $AB$ respectively, in such a way that $BN = 3NC$ and $AM = 2MB$ and segments $AN$ and $DM$ are drawn that intersect at point $P$, find the ratio between the areas of triangle $APM$ and trapezoid $ABCD$.

By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?

Let $S$ be the sum of all products $ab$ where $a$ and $b$ are distinct elements of the set $\{1,2,...,46\}$. Prove that $47$ divides $S$.

In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?

Is it possible to divide a polygon with $21$ sides into $2022$ triangles in such a way that among all the vertices there are not three collinear?