Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.

In Cartesian space, let $\Omega = \{(a, b, c) : a, b, c$ are integers between $1$ and $30\}$. A point of $\Omega$ is said to be visible from the origin if the segment that joins said point with the origin does not contain any other elements of $\Omega$. Find the number of points of $\Omega$ that are visible from the origin.

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. $1011$ points $P_1$, $P_2$, $P_3$, $...$, $P_{1011}$ on the side $AC$ and $1011$ points $Q_1$, $Q_2$, $Q_3$, $...$ ,$ Q_{1011}$ on side AB (see figure) in such a way as to generate $2023$ triangles of equal area. Find the length of the segment $AP_{1011}$. PS. Harder version of 2023 Chile NMO L1 P3

Inside a square with side $60$, $121$ points are drawn. Prove them are three points that are vertices of a triangle of area not exceeding $30$.

What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?

Let $\vartriangle ABC$ be a triangle such that $\angle ABC = 30^o$, $\angle ACB = 15^o$. Let $M$ be midpoint of segment $BC$ and point $N$ lies on segment $MC$, such that the length of $NC$ is equal to length of $AB$. Proce that $AN$ is the bisector of the angle $\angle MAC$.