The circle $\omega_1$ with diameter $[AB]$ and the circle $\omega_2$ with center $A$ intersects at points $C$ and $D$. Let $E$ be a point on the circle $\omega_2$, which is outside $\omega_1$ and at the same side as $C$ with respect to the line $AB$. Let the second point of intersection of the line $BE$ with $\omega_2$ be $F$. For a point $K$ on the circle $\omega_1$ which is on the same side as $A$ with respect to the diameter of $\omega_1$ passing through $C$ we have $2\cdot CK \cdot AC = CE \cdot AB$. Let the second point of intersection of the line $KF$ with $\omega_1$ be $L$. Show that the symmetric of the point $D$ with respect to the line $BE$ is on the circumcircle of the triangle $LFC$.

Let $m$ be a positive integer. a. Show that there exist infinitely many positive integers $k$ such that $1+km^3$ is a perfect cube and $1+kn^3$ is not a perfect cube for all positive integers $n<m$. b. Let $m=p^r$ where $p \equiv 2 \pmod 3$ is a prime number and $r$ is a positive integer. Find all numbers $k$ satisfying the condition in part a.

Let $G$ be a simple, undirected, connected graph with $100$ vertices and $2013$ edges. It is given that there exist two vertices $A$ and $B$ such that it is not possible to reach $A$ from $B$ using one or two edges. We color all edges using $n$ colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of $n$.

Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

Find the maximum value of $M$ for which for all positive real numbers $a, b, c$ we have \[ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) \]

Let $n$ be a positive integer and $P_1, P_2, \ldots, P_n$ be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances $P_iP_1, P_iP_2, \ldots, P_iP_n$ forms the same sequence for all $i=1,2, \ldots, n$ when these numbers are arranged in a non-decreasing order. Find all possible values of $n$.