Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $ABCD$ be a quadrilateral whose diagonals are perpendicular, and let $S$ be the intersection of those diagonals. Let $K, L, M$ and $N$ be the reflections of $S$ on the sides $AB$, $BC$, $CD$ and $DA$ respectively. $BN$ cuts the circumcircle of $\vartriangle SKN$ at $E$ and $BM$ cuts the circumcircle of $\vartriangle SLM$ at $F$. Prove that the quadrilateral $EFLK$ is cyclic.

Exactly $4n$ numbers in set $A= \{ 1,2,3,...,6n \} $ of natural numbers painted in red, all other in blue. Proved that exist $3n$ consecutive natural numbers from $A$, exactly $2n$ of which numbers is red.

Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively. If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.

Find the minimal number of cells on a $5\times 7$ board that must be painted so that any cell which is not painted has exactly one neighboring (having a common side) painted cell.

Define $a\clubsuit b=a^2b-ab^2$. Which of the following describes the set of points $(x, y)$ for which $x\clubsuit y=y\clubsuit x$? ${ \textbf{(A)}\ \text{a finite set of points} \\ \qquad\textbf{(B)}\ \text{one line} \\ \qquad\textbf{(C)}\ \text{two parallel lines}\\ \qquad\textbf{(D}}\ \text{two intersecting lines}\\ \qquad\textbf{(E)}\ \text{three lines} $

A city is a point on the plane. Suppose there are $n\geq 2$ cities. Suppose that for each city $X$, there is another city $N(X)$ that is strictly closer to $X$ than all the other cities. The government builds a road connecting each city $X$ and its $N(X)$; no other roads have been built. Suppose we know that, starting from any city, we can reach any other city through a series of road. We call a city $Y$ [i]suburban[/i] if it is $N(X)$ for some city $X$. Show that there are at least $(n-2)/4$ suburban cities. [i]Proposed by usjl.[/i]

Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$. Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$. a) Prove that: For all $m_1,m_2\in \mathbb{Z^+}$, we have:$$k([m_1,m_2])=[k(m_1),k(m_2)].$$(Here $[a,b]$ is the least common multiple of $a,b.$) b) Determine $k(2),k(4),k(5),k(10).$

There are $n$ different points $A_1, \ldots , A_n$ in the plain and each point $A_i$ it is assigned a real number $\lambda_i$ distinct from zero in such way that $(\overline{A_i A_j})^2 = \lambda_i + \lambda_j$ for all the $i$,$j$ with $i\neq{}j$} Show that: (1) $n \leq 4$ (2) If $n=4$, then $\frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3}+ \frac{1}{\lambda_4} = 0$

Every officially published book used to have an ISBN code (International Standard Book Number) which consisted of $10$ symbols. Such code looked like this: $$a_1a_2 . . . a_9a_{10}$$ with $a_1, . . . , a_9 \in \{0, 1, . . . , 9\}$ and $a_{10} \in \{0, 1, . . . , 9, X\}$. The symbol $X$ stood for the number $10$. With a valid ISBN code was $$a_1 + 2a2 + . . . + 9a_9 + 10a_{10}$$ a multiple of $11$. Prove the following statements. (a) If one symbol is changed in a valid ISBN code, the result is no valid ISBN code. (b) When two different symbols swap places in a valid ISBN code then the result is not a valid ISBN.

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ?

Sally's salary in 2006 was $\$37,500$. For 2007 she got a salary increase of $x$ percent. For 2008 she got another salary increase of $x$ percent. For 2009 she got a salary decrease of $2x$ percent. Her 2009 salary is $\$34,825$. Suppose instead, Sally had gotten a $2x$ percent salary decrease for 2007, an $x$ percent salary increase for 2008, and an $x$ percent salary increase for 2009. What would her 2009 salary be then?

Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$). Then Arnaldo tells the number of divisors of $ab$. Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.

Let $\mathbb{Z^+}$ and $\mathbb{P}$ denote the set of positive integers and the set of prime numbers, respectively. A set $A$ is called $S-\text{proper}$ where $A, S \subset \mathbb{Z^+}$ if there exists a positive integer $N$ such that for all $a \in A$ and for all $0 \leq b <a$ there exist $s_1, s_2, \ldots, s_n \in S$ satisfying $ b \equiv s_1+s_2+\cdots+s_n \pmod a$ and $1 \leq n \leq N.$ Find a subset $S$ of $\mathbb{Z^+}$ for which $\mathbb{P}$ is $S-\text{proper}$ but $\mathbb{Z^+}$ is not.

Prove that the numbers $\tan^2 20^o, \tan^2 40^o,\tan^2 80^o$ are the roots of the equation $x^3 - 33x^2 + 27x - 33 = 0$.

In the pyramid $SA_1A_2 \cdots A_n$, all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$. Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.

We call a coloring of an $m\times n$ table ($m,n\ge 5$) in three colors a [i]good coloring[/i] if the following conditions are satisfied: 1) Each cell has the same number of neighboring cells of two other colors; 2) Each corner has no neighboring cells of its color. Find all pairs $(m,n)$ ($m,n\ge 5$) for which there exists a good coloring of $m\times n$ table. [i](I. Manzhulina, B. Rubliov)[/i]

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Find all $3$-digit numbers $\overline {abc}$ such that $\overline {abc} = a! + b! + c! $.

A rubber band of negligible thickness encloses three pegs that lie in a perfect line, as shown. Each peg has a diameter of $4$ cm, as shown. What is the length of the rubber band used, in centimeters? All pegs shown are congruent circles. [asy] size(120); draw(circle((0,0),1));draw(circle((0,2),1));draw(circle((0,4),1)); dot((0,0)^^(0,2)^^(0,4)); draw((-1,0)--(-1,4)--arc((0,4),1,180,0)--(1,4)--(1,0)--arc((0,0),1,360,180),linewidth(2)); draw((-1,0)--(1,0),dotted); label("$4$ cm", (-0.38,0)--(1,0), N); [/asy] $\textbf{(A) }8\qquad\textbf{(B) }8+4\pi\qquad\textbf{(C) }16+4\pi\qquad\textbf{(D) }16+8\pi\qquad\textbf{(E) }16\pi$

A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$. We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$. Determine the greatest value of $n$.

Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$ $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$ $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$ converges to a limit and find these limits.

Let's call a natural number [i] interesting[/i] if any of its two digits consecutive forms a number that is a multiple of $19$ or $21$. For example, The number $7638$ is interesting, because $76$ is a multiple of $19$, $63$ is multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers of $2022$ digits exist?

300 students participate on the international math olympiad. Every student speaks in exactly two of the official languages of the olympiad and every language is spoken by 100 people (it is known that students speak only the official languages). Prove that the students can be sited on a circular table, such that no two neighbors spoke the same language.