We are given a non-infinite sequence $a_1,a_2…a_n$ of natural numbers. While it is possible, on each turn are chosen two arbitrary indexes $i<j$ such that $a_i \nmid a_j$, and then $a_i$ and $a_j$ are changed with their $gcd$ and $lcm$. Prove that this process is non-infinite and the created sequence doesn’t depend on the made choices.

Let $ a<a'<b<b'$ be real numbers and let the real function $ f$ be continuous on the interval $ [a,b']$ and differentiable in its interior. Prove that there exist $ c \in (a,b), c'\in (a',b')$ such that \[ f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a),\] \[ f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'),\] and $ c<c'$. [i]B. Szokefalvi Nagy[/i]

Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.

Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon. [asy] import geometry; size(10cm); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("$\omega_1$",8*dir(110),SW); label("$\omega_2$",5*dir(70)+(15,0),SE); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$B$",B,N+1/2*E); label("$A$",A,N+1/2*W); label("$C$",C,S+1/4*W); label("$D$",D,S+1/4*E); label("$15$",midpoint(O1--O2),N); label("$16$",midpoint(C--D),N); label("$2$",midpoint(A--B),S); label("$\Omega$",o.C+(o.r-1)*dir(270)); [/asy]

Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon? [asy] size(100); draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0)); draw((2,2)--(2,3)--(3,3)--(3,2)--cycle); [/asy]

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.

Prove that for every integer $S\ge100$ there exists an integer $P$ for which the following story could hold true: The mathematician asks the shop owner: ``How much are the table, the cabinet and the bookshelf?'' The shop owner replies: ``Each item costs a positive integer amount of Euros. The table is more expensive than the cabinet, and the cabinet is more expensive than the bookshelf. The sum of the three prices is $S$ and their product is $P$.'' The mathematician thinks and complains: ``This is not enough information to determine the three prices!'' (Proposed by Gerhard Woeginger, Austria)

At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average how many minutes can a student spend on a problem?

A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

We call an $n\times n$ matrix [i]well groomed[/i] if it only contains elements $0$ and $1$, and it does not contain the submatrix $\begin{pmatrix} 1& 0\\ 0 & 1 \end{pmatrix}.$ Show that there exists a constant $c>0$ such that every well groomed, $n\times n$ matrix contains a submatrix of size at least $cn\times cn$ such that all of the elements of the submatrix are equal. (A well groomed matrix may contain the submatrix $\begin{pmatrix} 0& 1\\ 1 & 0 \end{pmatrix}.$ )

There are some segments on the plane such that no two of them intersect each other (even at the ending points). We say segment $AB$ [b]breaks[/b] segment $CD$ if the extension of $AB$ cuts $CD$ at some point between $C$ and $D$. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(4cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -5.267474904743955, xmax = 11.572179069738377, ymin = -10.642621257034536, ymax = 4.543526642434019; /* image dimensions */ /* draw figures */ draw((-4,-2)--(1.08,-2.03), linewidth(2)); draw(shift((-2.1866176795507295,-2.0107089507113147))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((-0.16981767035094117,3.225314210196242)--(-2.1866176795507295,-2.0107089507113147), linewidth(2) + linetype("4 4")); draw((-0.16981767035094117,3.225314210196242)--(-0.8194002739586808,1.538865607509914), linewidth(2)); label("$A$",(-1.2684397405642523,3.860690076971137),SE*labelscalefactor,fontsize(16)); label("$B$",(-1.9211395070170559,2.002590777612728),SE*labelscalefactor,fontsize(16)); label("$C$",(-4.971261820527631,-1.6571211388676117),SE*labelscalefactor,fontsize(16)); label("$D$",(1.08925640451367566,-1.6571211388676117),SE*labelscalefactor,fontsize(16)); /* dots and labels */ dot((-4,-2),dotstyle); dot((1.08,-2.03),dotstyle); dot((-0.16981767035094117,3.225314210196242),dotstyle); dot((-0.8194002739586808,1.538865607509914),dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] $a)$ Is it possible that each segment when extended from both ends, breaks exactly one other segment from each way? [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(4cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.8, xmax = 8.68, ymin = -10.32, ymax = 3.64; /* image dimensions */ /* draw figures */ draw((-2.56,1.24)--(-0.36,1.4), linewidth(2)); draw((-3.32,-2.68)--(-1.24,-3.08), linewidth(2)); draw(shift((-2.551651190956802,-2.8277593863544612))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-0.8889576602618603,1.3615303519809556))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((-2.551651190956802,-2.8277593863544612)--(-0.8889576602618603,1.3615303519809556), linewidth(2) + linetype("4 4")); draw((-1.4097008194020806,0.049476186483185636)--(-1.8514772275312024,-1.0636149148218605), linewidth(2)); /* dots and labels */ dot((-2.56,1.24),dotstyle); dot((-0.36,1.4),dotstyle); dot((-3.32,-2.68),dotstyle); dot((-1.24,-3.08),dotstyle); dot((-1.4097008194020806,0.049476186483185636),dotstyle); dot((-1.8514772275312024,-1.0636149148218605),dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] $b)$ A segment is called [b]surrounded[/b] if from both sides of it, there is exactly one segment that breaks it.\\ ([i]e.g.[/i] segment $AB$ in the figure.) Is it possible to have all segments to be surrounded? [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(7cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -10.70976151557872, xmax = 18.64292748469251, ymin = -16.354300717041443, ymax = 9.136192362141452; /* image dimensions */ /* draw figures */ draw((1.0313140845297686,0.748205038977829)--(-1.3,-4), linewidth(2.8)); draw((-5.780195085389632,-2.13088646583346)--(-2.549994860479401,-2.13088646583346), linewidth(2.8)); draw((4.121070821400425,-3.816208322308361)--(1.78,-1.88), linewidth(2.8)); draw(shift((-0.38228674372374466,-2.13088646583346))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((-2.549994860479401,-2.13088646583346)--(-0.38228674372374466,-2.13088646583346), linewidth(2.8) + linetype("4 4")); draw(shift((0.32979226045261084,-0.6805897691262632))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((4.121070821400425,-3.816208322308361)--(0.32979226045261084,-0.6805897691262632), linewidth(2.8) + linetype("4 4")); draw((-3.6313140845297687,-8.74820503897783)--(3.600422205681574,5.980726991931396), linewidth(2.8) + linetype("2 2")); label("$A$",(-0.397698406272906,1.754593418658662),SE*labelscalefactor,fontsize(16)); label("$B$",(-2.6377720405041316,-3.266261278756151),SE*labelscalefactor,fontsize(16)); /* dots and labels */ dot((1.0313140845297686,0.748205038977829),linewidth(6pt) + dotstyle); dot((-1.3,-4),linewidth(6pt) + dotstyle); dot((-5.780195085389632,-2.13088646583346),linewidth(6pt) + dotstyle); dot((-2.549994860479401,-2.13088646583346),linewidth(6pt) + dotstyle); dot((4.121070821400425,-3.816208322308361),linewidth(6pt) + dotstyle); dot((1.78,-1.88),linewidth(6pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] [i]Proposed by Morteza Saghafian[/i]

On a line are given $n>3$ points. Find the number of colorings of these points in red and blue, such that in any set of consequent points the difference between the numbers of red and blue points does not exceed $2$.

Let $P$ be a point outside a circle centered at $O$. From $P$, tangent lines are drawn to the circle, touching the circle at points $A$ and $B$. Ray $\overrightarrow{BO}$ is drawn intersecting the circle again at $C$ and intersecting ray $\overrightarrow{PA}$ at $Q$. If $3QA = 2AP$, what is the value of $\sin \angle CAQ$?

Prove that the number $\underbrace{111\ldots 11}_{1997}\underbrace{22\ldots 22}_{1998}5$ (which has 1997 of 1-s and 1998 of 2-s) is a perfect square.

Determine all quadruples $(a, b, c, d)$ of real numbers satisfying the equation \[256a^3b^3c^3d^3 = (a^6+b^2+c^2+d^2)(a^2+b^6+c^2+d^2)(a^2+b^2+c^6+d^2)(a^2+b^2+c^2+d^6).\]

For all $x \geq 2$, $y \geq 2$ real numbers, prove that $$x(\frac{4x}{y-1}+\frac{1}{2y+x})+y(\frac{y}{6x-9}+\frac{1}{2x+y}) > \frac{26}{3}$$

On a piece of paper, we write down all positive integers $n$ such that all proper divisors of $n$ are less than $18$. We know that the sum of all numbers on the paper having exactly one proper divisor is $666$. What is the sum of all numbers on the paper having exactly two proper divisors? We say that $k$ is a [i]proper divisor [/i]of the positive integer $n$ if $k | n$ and $1 < k < n$.

Find all pairs $(a,b)$ of positive integers such that $a^2\mid b^3+1$ and $b^2\mid a^3+1$. [i]Linus Tang[/i]

Let $ a_1,a_2,...,a_n$ is permutaion of $ 1,2,...,n$. For this permutaion call the pair $ (a_i,a_j)$ [i]wrong pair [/i]if $ i<j$ and $ a_i >a_j$.Let [i]number of inversion [/i] is number of [i]wrong pair [/i] of permutation $ a_1,a_2,a_3,..,a_n$. Let $ n \ge 2$ is positive integer. Find the number of permutation of $ 1,2,..,n$ such that its [i]number of inversion [/i]is divisible by $ n$.

For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$.

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

For any positive integer $n{}$ define $a_n=\{n/s(n)\}$ where $s(\cdot)$ denotes the sum of the digits and $\{\cdot\}$ denotes the fractional part.[list=a] [*]Prove that there exist infinitely many positive integers $n$ such that $a_n=1/2.$ [*]Determine the smallest positive integer $n$ such that $a_n=1/6.$ [/list] [i]Marius Burtea[/i]

There is a balance without weights and there are two piles of stones of unknown masses, 10 stones in each pile. One is allowed an unlimited number of weighing iterations, but only 9 stones at most fit on any plate of the balance. Is it always possible to determine which stone pile is heavier or establish that they are equal? Sergey Dorichenko