$2022$ points are arranged in a circle, one of which is colored in black, and others in white. In one operation, The Hedgehog can do one of the following actions: 1) Choose two adjacent points of the same color and flip the color of both of them (white becomes black, black becomes white) 2) Choose two points of opposite colors with exactly one point in between them, and flip the color of both of them Is it possible to achieve a configuration where each point has a color opposite to its initial color with these operations? [i](Proposed by Oleksii Masalitin)[/i]

Points $ B_1$ and $ B_2$ lie on ray $ AM$, and points $ C_1$ and $ C_2$ lie on ray $ AK$. The circle with center $ O$ is inscribed into triangles $ AB_1C_1$ and $ AB_2C_2$. Prove that the angles $ B_1OB_2$ and $ C_1OC_2$ are equal.

Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$ is integer for every integer $d\ge 0$

$ABC$ triangle with $|AB|<|BC|<|CA|$ has the incenter $I$. The orthocenters of triangles $IBC, IAC$ and $IAB$ are $H_A, H_A$ and $H_A$. $H_BH_C$ intersect $BC$ at $K_A$ and perpendicular line from $I$ to $H_BH_B$ intersect $BC$ at $L_A$. $K_B, L_B, K_C, L_C$ are defined similarly. Prove that $$|K_AL_A|=|K_BL_B|+|K_CL_C|$$

How many primes less than $100$ have $7$ as the ones digit? (Assume the usual base ten representation) $\text{(A)} \ 4 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 7 \qquad \text{(E)} \ 8$

We say that a table with three rows or infinite columns is [i]cool [/i] if it was filled with natural numbers, and also whenever the same number m appears in two or more different places in the table, the numbers that appear in the cells immediately below said places (when they exist) are equal. For example, the following table is cool: [img]https://cdn.artofproblemsolving.com/attachments/5/7/16583a6a9434fd2792a4df48a733226cf2f560.png[/img] For each of the following two tables, decide whether it is possible to fill in the empty cells before the resulting tables are cool, explaining how to do this, or why it is not possible to do this. In both tables from the fifth column, the number in the third line is two units greater than the number in the first line. [img]https://cdn.artofproblemsolving.com/attachments/8/a/56d2f05ea09555c39da88f09eb5901a57567f0.png[/img]

In the convex hexagon $ABCDEF$, the angles at the vertices $B$, $C$, $E$, and $F$ are equal. Moreover, the equality \[AB+DE=AF+CD\] holds. Prove that the line $AD$ and the bisectors of the segments $BC$ and $EF$ have a common point.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that any real numbers $x{}$ and $y{}$ satisfy \[f(xf(x)+f(y))=f(f(x^2))+y.\]

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

(a) Prove that for every natural number $k$, there are positive integers $a_1<a_2<\ldots <a_k$ such that $a_i-a_j$ divides $a_i$ for all $1\leq i, j\leq k, i\neq j$. (b) Show that there is an absolute constant $C>0$ such that $a_1>k^{Ck}$ for every sequence $a_1,\ldots, a_k$ of numbers that satisfy the above divisibility condition. [A. Balogh, I. Z. Ruzsa]