Let $n \geq 2$ be an integer. Find the minimum $k$ for which there exists a partition of $\{1, 2, . . . , k\}$ into $n$ subsets $X_1,X_2, \cdots , X_n$ such that the following condition holds: for any $i, j, 1 \leq i < j \leq n$, there exist $x_i \in X_1, x_j \in X_2$ such that $|x_i - x_j | = 1.$

A rectangle \( m \times n \), where \( m \) and \( n \) are natural numbers strictly greater than 1, is partitioned into \( mn \) unit squares, each of which can be colored either black or white. An operation consists of changing the color of all the squares in a row or in a column to the opposite color. Is it possible that, although initially exactly one square is colored black and all the others are white, after a finite number of moves all squares have the same color?

Denote $\langle x\rangle$ the distance of the real number x from the nearest integer. Let f be a linear, 1 periodic, continuous real function. Prove that there exist natural n and real numbers $a_1 , ..., a_n , b_1 , ..., b_n , c_1 , ..., c_n$ such that $$f(x) = \sum_{i = 1}^n c_i \langle a_ix + b_i \rangle$$ for every x iff there is a k such that $$\sum_{j = 1}^{2^k} f \left(x+{j\over2^k}\right)$$ is constant.

There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.

Given an infinite sequence of numbers $a_1, a_2, a_3, ...$ such that for each positive integer $k$, there exists positive integer $t$ for which $a_k = a_{k+t} = a_{k+2t} = ....$ Does this sequences must be periodic?

Prove that: $$\left\lfloor10^{n+3}\cdot\sqrt{\overline{\underbrace{11\ldots1}_{2n\text{ times}}}}\right\rfloor=\overline{\underbrace{33\ldots3}_{2n\text{ times}}166}$$ [i]Proposed by Ovidiu Pop[/i]

Calculate: $\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}$

We consider an acute triangle $ABC$. The altitude from $A$ is $AD$, the altitude from $D$ in triangle $ABD$ is $DE,$ and the altitude from $D$ in triangle $ACD$ is $DF$. a) Prove that the triangles $ABC$ and $AF E$ are similar. b) Prove that the segment $EF$ and the corresponding segments constructed from the vertices $B$ and $C$ all have the same length.

In triangle $ABC$, the perpendicular bisectors of sides $AB$ and $BC$ intersect side $AC$ at points $P$ and $Q$, respectively, with point $P$ lying on the segment $AQ$. Prove that the circumscribed circles of the triangles $PBC$ and $QBA$ intersect on the bisector of the angle $PBQ$.

Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?

show that $\frac1{15} < \frac12\cdot\frac34\cdots\frac{99}{100} < \frac1{10}$

Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when: a) $n=2014$ b) $n=2015 $ c) $n=2018$

Given a trapezoid $ABCD$ with bases $BC$ and $AD$, with $AD=2 BC$. Let $M$ be the midpoint of $AD, E$ be the intersection point of the sides $AB$ and $CD$, $O$ be the intersection point of $BM$ and $AC, N$ be the intersection point of $EO$ and $BC$. In what ratio, point $N$ divides the segment $BC$?

Let $n\geq1$ be a positive integer. $n$ lamps are placed in a line. At minute 0, some lamps are on (maybe all of them). Every minute the state of the lamps changes: A lamp is on at minute $t+1$ if and only if at minute $t$, exactly one of its neighbors is on (the two lamps at the ends have one neighbor each, all other lamps have two neighbors). For which values of $n$ can we guarantee that all lamps will be off after some time?

Let $a$ and $b$ be complex numbers such that $(a+1)(b+1)=2$ and $(a^2+1)(b^2+1)=32.$ Compute the sum of all possible values of $(a^4+1)(b^4+1).$ [i]Proposed by Kyle Lee[/i]

Seven different positive integers are written on a sheet of paper. The result of the multiplication of the seven numbers is the cube of a whole number. If the largest of the numbers written on the sheet is $N$, determine the smallest possible value of $N$. Show an example for that value of $N$ and explain why $N$ cannot be smaller.

Let $ \omega$ be a circle with center O and radius 10, and let H be a point such that $OH = 6$. A point P is called snug if, for all triangles ABC with circumcircle ω and orthocenter $H$, we have that P lies on $\triangle$ABC or in the interior of $\triangle$ABC. Find the area of the region consisting of all snug points.

For which $n \ge 2$ can one find a sequence of distinct positive integers $a_1, a_2, \ldots , a_n$ so that the sum $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots +\frac{a_n}{a_1}$$ is an integer? [i](3 points)[/i]

Let $ a,b,$ and $ c$ be digits with $ a\ne0.$ The three-digit integer $ abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $ acb$ lies two thirds of the way between the same two squares. What is $ a \plus{} b \plus{} c$? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 21$

Cyclic quadrilateral $ABCD$ with circumcircle $\omega$ is given. Let $E$ be a fixed point on segment $AC$. $M$ is an arbitrary point on $\omega$, lines $AM$ and $BD$ meet at a point $P$. $EP$ meets $AB$ and $AD$ at points $R$ and $Q$, respectively, $S$ is the intersection of $BQ,DR$ and lines $MS$ and $AC$ meet at a point $T$. Prove that as $M$ varies the circumcircle of triangle $\bigtriangleup CMT$ passes through a fixed point other than $C$. [i]Proposed by Chunlai Jin - China[/i]

Given $n>3$ points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) $3$ of the given points and not containing any other of the $n$ points in its interior ?

Let $a$ and $k$ be positive integers such that $a^2+k$ divides $(a-1)a(a+1)$. Prove that $k\ge a$.

Let $n$ be a positive integer and let $f_1(x), f_2(x) \dots f_n(x)$ be affine functions from $\mathbb{R}$ to $\mathbb{R}$ such that, amongst the graph of these functions, no two are parallel and no three are concurrent. Let $S$ be the set of all convex functions $g(x)$ from $\mathbb{R}$ to $\mathbb{R}$ such that for each $x \in \mathbb{R}$, there exists $i$ such that $g(x) = f_i(x)$. Determine the largest and smallest possible values of $|S|$ in terms of $n$. (A function $f(x)$ is affine if it is of form $f(x) = ax + b$ for some $a, b \in \mathbb{R}$. A function $g(x)$ is convex if $g(\lambda x + (1 - \lambda) y) \leq \lambda g(x) + (1-\lambda)g(y)$ for all $x, y \in \mathbb{R}$ and $0 \leq \lambda \leq 1$)

Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le p(1) \le 10^7$. Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$. Determine all possible values of $p(1)$. (Note: $1999$ is a prime number.)