All the squares of a $2022 \times 2022$ board will be colored white or black. Chips will be placed in several of these boxes, at most one per box. We say that two tokens attack each other, when the following two conditions are met: a) There is a path of squares that joins the squares where the pieces were placed. This path can have a horizontal, vertical, or diagonal direction. b) All the squares in this path, including the squares where the pieces are, are of the same color. For example, the following figure shows a small example of a possible coloring of a $6 \times 6$ board with $A, B, C, D$, and $E$ tiles placed. The pairs of checkers that attack each other are $(D, E)$, $(C, D)$, and $(B, E)$. [img]https://cdn.artofproblemsolving.com/attachments/2/0/52ec7b7d1c02e266b666e4f8b25e87c58f0c89.png[/img] What is the maximum value of $k$ such that it is possible to color the board and place $k$ tiles without any two of them attacking each other?

Let $k$ be the smallest positive integer for which there exist distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial $$p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)$$ has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ for which this minimum $k$ is achieved.

Find all prime numbers $p$ such that the number $$3^p+4^p+5^p+9^p-98$$ has at most $6$ positive divisors.

In a company of people some pairs are enemies. A group of people is called [i]unsociable[/i] if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part. [i]Proposed by Russia[/i]

On the sides of the triangle $ABC$ externally constructed right triangles $ABC_1$, $BCA_1$, $CAB_1$. Prove that the points of intersection of the medians of the triangles $ABC$ and $A_1B_1C_1$ coincide.

In an acute triangle $ABC$, the incircle $\omega$ touches $BC$ at $D$. Let $I_a$ be the excenter of $ABC$ opposite to $A$, and let $M$ be the midpoint of $DI_a$. Prove that the circumcircle of triangle $BMC$ is tangent to $\omega$. [i]Patrik Bak (Slovakia)[/i]

Given is a triangle $ABC$ and a circle through $A, B$. The perpendicular bisector of $AB$ meets the circle at $P, Q$, such that $AP>AQ$. Let $M$ be a point on the segment $AB$. The lines through $M$, parallel to $QA, QB$ meet $PB, PA$ at $R, S$. Prove that $MQ$ bisects $RS$.

$ABCD$ is a quadrilateral on the complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a $\textbf{(A)}~\text{Rectangle}$ $\textbf{(B)}~\text{Rhombus}$ $\textbf{(C)}~\text{Isosceles Trapezium}$ $\textbf{(D)}~\text{Square}$

Let $A$ be a set of $8$ elements, and $B := (B_1,...,B_7)$ be an ordered $7$-tuple of subsets of $A$. Let $N$ be the number of such $7$-tuples $B$ such that there exists a unique $4$-element subset $I \subseteq \{1,2,...,7\}$ for which the intersection $\cap _{ i\in I} B_i$ is nonempty. Find the remainder when $N$ is divided by $67$.

Cut the $10$ cm $\times 20$ cm rectangle into two pieces with one straight cut so that they can fit inside the $19.5$ cm diameter circle without intersecting.

A sphere touches all edges of a tetrahedron. Let $a, b, c$ and d be the segments of the tangents to the sphere from the vertices of the tetrahedron. Is it true that that some of these segments necessarily form a triangle? (It is not obligatory to use all segments. The side of the triangle can be formed by two segments)

Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

Let $b$ be the probability that the cards are from different suits. Compute $\lfloor1000b\rfloor$.

Non-negative real numbers $x, y, z$ satisfy the following relations: $3x + 5y + 7z = 10$ and $x + 2y + 5z = 6$. Find the minimum and maximum of $w = 2x - 3y + 4z$.

Find all ordered triples $(a,b,c)$ of positive integers such that the value of the expression \[\left (b-\frac{1}{a}\right )\left (c-\frac{1}{b}\right )\left (a-\frac{1}{c}\right )\] is an integer.

Find all pairs of positive integers $ (a, b)$ such that \[ ab \equal{} gcd(a, b) \plus{} lcm(a, b). \]

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

Let $ P$ be an interior point of an acute angled triangle $ ABC$. The lines $ AP,BP,CP$ meet $ BC,CA,AB$ at points $ D,E,F$ respectively. Given that triangle $ \triangle DEF$ and $ \triangle ABC$ are similar, prove that $ P$ is the centroid of $ \triangle ABC$.

Determine the smallest integer $n \ge 2012$ for which it is possible to have $16$ consecutive integers on a $4 \times 4$ board so that, if we select $4$ elements of which there are not two in the same row or in the same column, the sum of them is always equal to $n$. For the $n$ found, show how to fill the board.

$n$ numbers are written on a blackboard. Someone then repeatedly erases two numbers and writes half their arithmetic mean instead, until only a single number remains. If all the original numbers were $1$, show that the final number is not less than $\frac{1}{n}$.

We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]?

Write $0.2010\overline{228}$ as a fraction.

There are 32 people at a conference. Initially nobody at the conference knows the name of anyone else. The conference holds several 16-person meetings in succession, in which each person at the meeting learns (or relearns) the name of the other fifteen people. What is the minimum number of meetings needed until every person knows everyone else's name? [i]David Yang, Victor Wang.[/i] [size=85][i]See the "odd version" [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=810&t=500914]here[/url].[/i][/size]

For an $A=\{ a_i\}^{\infty}_{i=0}$ sequence let $SA=\{ a_0, a_0+a_1, a_0+a_1+a_2, \ldots\}$ be the sequence of partial sums of the $a_0+a_1+\ldots$ series. Does there exist a non-identically zero sequence $A$ such that all of the sequences $A, SA, SSA, SSSA, \ldots$ are convergent? (translated by Miklós Maróti)

Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, while $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers where $c$ is not divisible by the square of any prime. Compute $a+b+c$. [i]Proposed by Andrew Wen[/i]