All cells of an $n\times n$ table are painted in several colors so that there is no monochromatic $2\times2$ square. A sequence of different cells $a_1,a_2,\ldots,a_k$ is called a [i]colorful[/i] if any two consecutive cells are adjacent and are painted in different colors. What is the largest $k{}$ for which there is a colorful sequence of length $k{}$ regardless of the coloring of the cells of the table? [i]Proposed by N. Belukhov[/i]

In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.

Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$. (1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin. (2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized. 50 points

Let $I$ be the incenter of triangle $ABC$. The circle of centre $A$ and radius $AI$ intersects the circumcircle of triangle $ABC$ in $M$ and $N$. Prove that the line $MN$ is tangent to the incircle of triangle $ABC$

Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

A city is laid out with a rectangular grid of roads with 10 streets numbered from 1 to 10 running east-west and 16 avenues numbered from 1 to 16 running northsouth. All streets end at First and Sixteenth Avenues, and all avenues end at First and Tenth Streets. A rectangular city park is bounded on the north and south by Sixth Street and Eighth Street, and bounded on the east and west by Fourth Avenue and Twelfth Avenue. Although there are no breaks in the roads that bound the park, no road goes through the park. The city paints a crosswalk from every street corner across any adjacent road. Thus, where two roads cross such as at Second Street and Second Avenue, there are four crosswalks painted, while at corners such as First Street and First Avenue, there are only two crosswalks painted. How many crosswalks are there painted on the roads of this city?

Prove that in any described $8$-gon there is a side that does not exceed the diameter of the inscribed circle in length. [i]P.A.Kozhevnikov[/i]

Do there exist $2011$ positive integers $a_1 < a_2 < \ldots < a_{2011}$ such that $\gcd(a_i,a_j) = a_j - a_i$ for any $i$, $j$ such that $1 \le i < j \le 2011$?

Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.

Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the first $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?

[b]Q.[/b] Consider in the plane $n>3$ different points. These have the properties, that all $3$ points can be included in a triangle with maximum area $1$. Prove that all the $n>3$ points can be included in a triangle with maximum area $4$. [i]Proposed by TuZo[/i]

There are $n > 2022$ cities in the country. Some pairs of cities are connected with straight two-ways airlines. Call the set of the cities {\it unlucky}, if it is impossible to color the airlines between them in two colors without monochromatic triangle (i.e. three cities $A$, $B$, $C$ with the airlines $AB$, $AC$ and $BC$ of the same color). The set containing all the cities is unlucky. Is there always an unlucky set containing exactly 2022 cities?

Prove that the equation $2+x^3y+y^2+z^2=0$ has no solutions in integers.

$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $handshakes, what is $N$?

Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$. [i]B. Berceanu[/i]

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

Consider the set $E$ consisting of pairs of integers $(a, b)$, with $a \geq 1$ and $b \geq 1$, that satisfy in the decimal system the following properties: [b](i)[/b] $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$, $\alpha_2 \neq 0$; [b](ii)[/b] $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$; [b](iii)[/b] $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$ Find the elements of $E$.

Given a polynomial$$P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0\in \mathbb{Z}[x]$$ with degree $n\ge 2$ and $a_o\ne 0.$ Prove that if $|a_{n-1}|>1+|a_{n-2}|+...+|a_1|+|a_0|$, then $P(x)$ is irreducible in $\mathbb{Z}[x].$

Suppose $a_1 =\frac16$ and $a_n = a_{n-1} - \frac{1}{n}+ \frac{2}{n + 1} - \frac{1}{n + 2}$ for $n > 1$. Find $a_{100}$.

Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.

Evaluate the sum $1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 \cdots + 208 + 209 - 210.$

It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$, maximum number of separators

Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$. [i](Proposed by Fedir Yudin)[/i]

In year $ N$, the $ 300^\text{th}$ day of the year is a Tuesday. In year $ N \plus{} 1$, the $ 200^{\text{th}}$ day of the year is also a Tuesday. On what day of the week did the $ 100^\text{th}$ day of year $ N \minus{} 1$ occur? $ \textbf{(A)}\ \text{Thursday} \qquad \textbf{(B)}\ \text{Friday} \qquad \textbf{(C)}\ \text{Saturday} \qquad \textbf{(D)}\ \text{Sunday} \qquad \textbf{(E)}\ \text{Monday}$