Danielle draws a point $O$ on the plane and a set of points $\mathcal P = \{P_0, P_1, \ldots , P_{2022}\}$ such that $$\angle{P_0OP_1} = \angle{P_1OP_2} = \cdots = \angle{P_{2021}OP_{2022}} = \alpha, \hspace{5pt} 0 < \alpha < \pi,$$where the angles are measured counterclockwise and for $0 \leq n \leq 2022$, $OP_n = r^n$, where $r > 1$ is a given real number. Then, obtain new sets of points in the plane by iterating the following process: given a set of points $\{A_0, A_1, \ldots , A_n\}$ in the plane, it is built a new set of points $\{B_0, B_1, \ldots , B_{n-1}\}$ such that $A_kA_{k+1}B_k$ is an equilateral triangle oriented clockwise for $0 \leq k \leq n - 1$. After carrying out the process $2022$ times from the set $P$, Danielle obtains a single point $X$. If the distance from $X$ to point $O$ is $d$, show that $$(r-1)^{2022} \leq d \leq (r+1)^{2022}.$$

Let $a_1,a_2,\cdots,a_n>0$ $(n\geq 2)$. Prove that$$\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot min \{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i$$

$a,b,c,d$ are real numbers so that $a\geq b,c\geq d$,\[|a|+2|b|+3|c|+4|d|=1.\] Let $P=(a-b)(b-c)(c-d)$,find the maximum and minimum value of $P$.

Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.

Let $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$ be five different points which are inside $D$ or on the border of figure $D$. Let $M=min\left\{P_iP_j \mid i \neq j\right\}$ be minimal distance between different points $P_i$. For which configuration of points $P_i$, value $M$ is at maximum, if : $a)$ $D$ is unit square $b)$ $D$ is equilateral triangle with side equal $1$ $c)$ $D$ is unit circle, circle with radius $1$

There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.

There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.