Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$ It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+x_6=s_{m}$. Find all possible values of $m$.

Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]

Let $M$ be a closed, orientable $3$-dimensional differentiable manifold, and let $G$ be a finite group of orientation preserving diffeomorphisms of $M$. Let $P$ and $Q$ denote the set of those points of $M$ whose stabilizer is nontrivial (that is, contains a nonidentity element of $G$) and noncyclic, respectively. Let $\chi (P)$ denote the Euler characteristic of $P$. Prove that the order of $G$ divides $\chi (P)$, and $Q$ is the union of $-2\frac{\chi(P)}{|G|}$ orbits of $G$.

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.

The angles $A$ and $B$ of base of the isosceles triangle $ABC$ are equal to $40^o$. Inside $\vartriangle ABC$, $P$ is such that $\angle PAB = 30^o$ and $\angle PBA = 20^o$. Calculate, without table, $\angle PCA$.

Prove that there are only [b]finitely[/b] positive integer $a$ such that $a-2006=\sum\limits_{i=1}^{2006} 2^ia_i$ with $\{a_i\}$ as divisors (not necessary distinct) of $n$.

For a fixed natural number $m \geq 2$, prove that [b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\] [b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.

Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$ (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E) None of the above.

In triangle $ABC,$ we have $\angle A \leq 90^\circ$ and $\angle B = 2 \angle C.$ The interior bisector of the angle $C$ meets the median $AM$ in $D.$ Prove that $\angle MDC \leq 45^\circ.$ When does equality hold? [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqqqzz = rgb(0,0,0.6); pen ffqqtt = rgb(1,0,0.2); pen qqwuqq = rgb(0,0.39,0); draw((3.14,5.81)--(3.23,0.74),ffqqtt+linewidth(2pt)); draw((3.23,0.74)--(9.73,1.32),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.14,5.81),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.19,2.88)); draw((3.14,5.81)--(6.71,1.05)); pair parametricplot0_cus(real t){ return (0.7*cos(t)+5.8,0.7*sin(t)+2.26); } draw(graph(parametricplot0_cus,-0.9270442638657642,-0.23350086562377703)--(5.8,2.26)--cycle,qqwuqq); dot((3.14,5.81),ds); label("$A$", (2.93,6.09),NE*lsf); dot((3.23,0.74),ds); label("$B$", (2.88,0.34),NE*lsf); dot((9.73,1.32),ds); label("$C$", (10.11,1.04),NE*lsf); dot((3.19,2.88),ds); label("$X$", (2.77,2.94),NE*lsf); dot((6.71,1.05),ds); label("$M$", (6.75,0.5),NE*lsf); dot((5.8,2.26),ds); label("$D$", (5.89,2.4),NE*lsf); label("$\alpha$", (6.52,1.65),NE*lsf); clip((-3.26,-11.86)--(-3.26,8.36)--(20.76,8.36)--(20.76,-11.86)--cycle); [/asy]

Let ABC with orthocenter $H$ and circumcenter $O$ be an acute scalene triangle satisfying $AB = AM$ where $M$ is the midpoint of $BC$. Suppose $Q$ and $K$ are points on $(ABC)$ distinct from A satisfying $\angle AQH = 90$ and $\angle BAK = \angle CAM$. Let $N$ be the midpoint of $AH$. • Let $I$ be the intersection of $B\text{-midline}$ and $A\text{-altitude}$ Prove that $IN = IO$. • Prove that there is point $P$ on the symmedian lying on circle with center $B$ and radius $BM$ such that $(APN)$ is tangent to $AB$. [i]Proposed by Krutarth Shah[/i]

What is the value of $$\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?$$ $\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5$

A class of $10$ children is divided into $5$ pairs of partners. Each pair of partners sits next to each other and works together during class. One day, the teacher decides he wants to divide the class into two groups. In order to make sure the students work with new people, he makes sure not to put any student in the same group as his or her partner. How many different ways can he divide the class into these two groups? $\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }32$

Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality \[ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. \] Show that the following inequality holds for all positive integers $k,m$ \[ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. \]

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

The following game is played on a rectangular chessboard $ 5 \times 9$ (with five rows and nine columns). Initially, a number of discs are randomly placed on some of the squares of the chessboard, with at most one disc on each square. A complete move consists of the moving every disc subject to the following rules: $ (1)$ Each disc may be moved one square up, down, left or right; $ (2)$ If a particular disc is moved up or down as part of a complete move, then it must be moved left or right in the next complete move; $ (3)$ If a particular disc is moved left or right as part of a complete move, then it must be moved up or down in the next complete move; $ (4)$ At the end of a complete move, no two discs can be on the same square. The game stops if it becomes impossible to perform a complete move. Prove that if initially $ 33$ discs are placed on the board then the game must eventually stop. Prove also that it is possible to place $ 32$ discs on the boards in such a way that the game could go on forever.

A particle with charge $8.0 \, \mu\text{C}$ and mass $17 \, \text{g}$ enters a magnetic field of magnitude $\text{7.8 mT}$ perpendicular to its non-zero velocity. After 30 seconds, let the absolute value of the angle between its initial velocity and its current velocity, in radians, be $\theta$. Find $100\theta$. [i](B. Dejean, 5 points)[/i]

An odd prime $p$ is called a prime of the year $2022$ if there is a positive integer $n$ such that $p^{2022}$ divides $n^{2022}+2022$. Show that there are infinitely many primes of the year $2022$.

You have a $7\times 7$ board divided into $49$ boxes. Mateo places a coin in a box. a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the $48$ remaining squares, without gaps or overlaps, using $15$ $3\times1$ rectangles and a cubit of three squares, like those in the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/9/a467439094376cd95c6dfe3e2ad3e16fe9f124.png[/img] b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using $14$ $3\times1$ rectangles and two cubits of three squares.

Let $S_n$ be the area of the figure enclosed by a curve $y=x^2(1-x)^n\ (0\leq x\leq 1)$　and the $x$-axis. Find $\lim_{n\to\infty} \sum_{k=1}^n S_k.$

Drawing all diagonals in a regular $2013$-gon, the regular $2013$-gon is divided into non-overlapping polygons. Prove that there exist exactly one $2013$-gon out of all such polygons.

The touching point of the excircle with the side of a triangle and the base of the altitude to this side are symmetric wrt the base of the corresponding bisector. Prove that this side is equal to one third of the perimeter.

Points $K, L, M$ and $N$ lying on the sides $AB, BC, CD$ and $DA$ of a square $ABCD$ are vertices of another square. Lines $DK$ and $N M$ meet at point $E$, and lines $KC$ and $LM$ meet at point $F$ . Prove that $EF\parallel AB$.

َA natural number $n$ is given. Let $f(x,y)$ be a polynomial of degree less than $n$ such that for any positive integers $x,y\leq n, x+y \leq n+1$ the equality $f(x,y)=\frac{x}{y}$ holds. Find $f(0,0)$.

Prove that the number of pairs $ \left(\alpha,S\right)$ of a permutation $ \alpha$ of $ \{1,2,\dots,n\}$ and a subset $ S$ of $ \{1,2,\dots,n\}$ such that \[ \forall x\in S: \alpha(x)\not\in S\] is equal to $ n!F_{n \plus{} 1}$ in which $ F_n$ is the Fibonacci sequence such that $ F_1 \equal{} F_2 \equal{} 1$

Given a set $I=\{(x_1,x_2,x_3,x_4)|x_i\in\{1,2,\cdots,11\}\}$. $A\subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\in A$, there exists $i,j(1\leq i<j\leq4)$, $(x_i-x_j)(y_i-y_j)<0$. Find the maximum value of $|A|$.