(a) Prove that if $n$ is a integer such that $n \geq 4011^2$ then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \] (b) Find the smallest positive integer $M$ for which whenever an integer $n$ is such that $n \geq M$ then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \]

$\bigtriangleup ABC$ triangle such that $AB = AC, \angle BAC = 20 \textdegree$. $P$ is on $AB$ such that $AP = BC$, find $\frac{1}{2}\angle APC$ in degrees.

In each cell of a square table of size $n \times n$ cells ($n \ge 3$) the number $1$ or $-1$ is written. If you take any two lines, multiply numbers standing above each other in them and add the n resulting products, then the sum will be equal to $0$. Prove that the number $n$ is divisible by $4$.

$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$. [hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600[/hide]

On a connected graph $G$, one may perform the following operations: [list] [*]choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours [*] choose a vertice $v$ with odd degree and delete it [/list] Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique. Proposed by [i]idonthaveanaopsaccount[/i]

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point. What is the largest possible number of red points? [i]Proposed by Art Waeterschoot, Belgium[/i]

Let $ A,B,C,M$ points in the plane and no three of them are on a line. And let $ A',B',C'$ points such that $ MAC'B, MBA'C$ and $ MCB'A$ are parallelograms: (a) Show that \[ \overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} < \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.\] (b) Assume segments $ AA', BB'$ and $ CC'$ have the same length. Show that $ 2 \left(\overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} \right) \leq \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.$ When do we have equality?

How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?

Lucky and Jinx were given a paper with $2023$ points arranged as the vertices of a regular polygon. They were then tasked to color all the segments connecting these points such that no triangle formed with these points has all edges in the same color, nor in three different colors and no quadrilateral (not necessarily convex) has all edges in the same color. After the coloring it was determined that Jinx used at least two more colors than Lucky. How many colors did each of them use? [i]Authored by Ilija Jovcheski[/i]

The roots of $ (x^2\minus{}3x\plus{}2)(x)(x\minus{}4)\equal{}0$ are: $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 0\text{ and }4 \qquad \textbf{(C)}\ 1\text{ and }2 \qquad \textbf{(D)}\ 0,1,2\text{ and }4\qquad \textbf{(E)}\ 1,2\text{ and }4$

Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$. Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$.

Let $a$ and $b$ be non-negative integers. Consider a sequence $s_1$, $s_2$, $s_3$, $. . .$ such that $s_1 = a$, $s_2 = b$, and $s_{i+1} = |s_i - s_{i-1}|$ for $i \ge 2$. Prove that there is some $i$ for which $s_i = 0$.

In an equilateral triangle $ABC$, let $D$ be a point on the side $AB$ such that $AD = AB /n$. Prove that the sum of $n - 1$ angles $\angle DP_lA$, $\angle DP_2A$, $...$, $\angle DP_nA$ where $P_1$, $P_2$, $...$ ,$P_{n-1}$ are the points dividing the side $BC$ into $n$ equal parts, is equal to $30$ degrees if (a) $n = 3$ (b) $n$ is an arbitrary integer, $n > 2$. (V Proizvolov)

Let $ ABC$ be a triangle where $ \angle ACB\equal{}90^{\circ}$. Let $ D$ be the midpoint of $ BC$ and let $ E$, and $ F$ be points on $ AC$ such that $ CF\equal{}FE\equal{}EA$. The altitude from $ C$ to the hypotenuse $ AB$ is $ CG$, and the circumcentre of triangle $ AEG$ is $ H$. Prove that the triangles $ ABC$ and $ HDF$ are similar.

Find all integer solutions of $ p^3\equal{}p^2\plus{}q^2\plus{}r^2$ where $ p,q,r$ are primes.

Number of solutions of the equation $3^x+4^x=8^x$ in reals is $\textbf{(A)}~0$ $\textbf{(B)}~1$ $\textbf{(C)}~2$ $\textbf{(D)}~\infty$

There are red and blue balls in an urn : $1024$ in total. In one round, we do the following: we draw the balls from the urn two by two. After all balls have been drawn, we put a new ball back into the urn for each pair of drawn balls: the colour of the new ball depends on that of the drawn pair. For two red balls drawn, we put back a red ball. For two blue balls, we put back a blue ball. For a red and a blue ball, we put back a black ball. For a red and a black ball, we put back a red ball. For a blue and a black ball, we put back a blue ball. Finally, for two black balls we put back a black ball. Then the next round begins. After $10$ rounds, a single ball remains in the urn, which is red. What is the maximal number of blue balls that might have been in the urn at the very beginning?

Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$, where $n$ is a positive integer.

A line is drawn through the $A$ vertex of triangle $ABC$ with $|AB|\ne|AC|$. Prove that the line can not contain more than one point $M$ such, that $M$ is not a triangle vertex, and $\angle ABM = \angle ACM$. What lines do not contain such a point $M$ at all?

Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$. [i]Proposed by Iurie Boreico[/i]

In an acute triangle $ABC$, let $M$ and $N$ be the midpoints of $AB$ and $AC$ and let $BH$ be its altitude from $B$. Its incircle touches $AC$ at $K$ and the line through $K$ parallel to $MH$ meets $MN$ at $P$. Prove that $AMPK$ has an incircle.

Let $X,Y$ be two points which lies on the line $BC$ of $\triangle ABC(X,B,C,Y\text{lies in sequence})$ such that $BX\cdot AC=CY\cdot AB$, $O_1,O_2$ are the circumcenters of $\triangle ACX,\triangle ABY$, $O_1O_2\cap AB=U,O_1O_2\cap AC=V$. Prove that $\triangle AUV$ is a isosceles triangle.

In triangle $ABC$ the incenter is $I$. The center of the excircle opposite $A$ is $I_A$, and it is tangent to $BC$ at $D$. The midpoint of arc $BAC$ is $N$, and $NI$ intersects $(ABC)$ again at $T$. The center of $(AID)$ is $K$. Prove that $TI_A\perp KI$.

Find all sets $S$ of positive integers that satisfy all of the following. $1.$ If $a,b$ are two not necessarily distinct elements in $S$, then $\gcd(a,b)$, $ab$ are also in $S$. $2.$ If $m,n$ are two positive integers with $n\nmid m$, then there exists an element $s$ in $S$ such that $m^2\mid s$ and $n^2\nmid s$. $3.$ For any odd prime $p$, the set formed by moduloing all elements in $S$ by $p$ has size exactly $\frac{p+1}2$.