Let $n$ be a positive integer. (a) Prove that there exists a set $S$ of $6n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32n^2$. (b) Prove that every set $T$ of $6n$ pairwise different positive integers contains two elements the least common multiple of which is larger than $9n^2$.

There were $n\ge2$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ all the teams could end up with the same number of points?

Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{1}{a}+\frac{3}{b}+\frac{5}{c} \ge 4a^2 + 3b^2 + 2c^2$$ When does the equality hold? Marius Stanean

Sketch the graph of $u(x, 1)$, if $0 \le x\le1$, \[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2\]

In an acute triangle $ABC$ with $AC > AB$, let $D$ be the projection of $A$ on $BC$, and let $E$ and $F$ be the projections of $D$ on $AB$ and $AC$, respectively. Let $G$ be the intersection point of the lines $AD$ and $EF$. Let $H$ be the second intersection point of the line $AD$ and the circumcircle of triangle $ABC$. Prove that \[AG \cdot AH=AD^2\]

Bas has coloured each of the positive integers. He had several colours at his disposal. His colouring satises the following requirements: • each odd integer is coloured blue, • each integer $n$ has the same colour as $4n$, • each integer $n$ has the same colour as at least one of the integers $n+2$ and $n + 4$. Prove that Bas has coloured all integers blue.

Six-hundred fifty students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti? [asy] size(200); defaultpen(linewidth(0.7)); defaultpen(fontsize(8)); draw(origin--(0,250)); int i; for(i=0; i<6; i=i+1) { draw((0,50*i)--(5,50*i)); } filldraw((25,0)--(75,0)--(75,150)--(25,150)--cycle, gray, black); filldraw((75,0)--(125,0)--(125,100)--(75,100)--cycle, gray, black); filldraw((125,0)--(175,0)--(175,150)--(125,150)--cycle, gray, black); filldraw((225,0)--(175,0)--(175,250)--(225,250)--cycle, gray, black); label("$50$", (0,50), W); label("$100$", (0,100), W); label("$150$", (0,150), W); label("$200$", (0,200), W); label("$250$", (0,250), W); label(rotate(90)*"Lasagna", (50,0), S); label(rotate(90)*"Manicotti", (100,0), S); label(rotate(90)*"Ravioli", (150,0), S); label(rotate(90)*"Spaghetti", (200,0), S); label(rotate(90)*"$\mbox{Number of People}$", (-40,140), W);[/asy] $\textbf{(A)} \: \frac25\qquad \textbf{(B)} \: \frac12\qquad \textbf{(C)} \: \frac54\qquad \textbf{(D)} \: \frac53\qquad \textbf{(E)} \: \frac52$

$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square. Prove $a$ is one of $a_1,\dots ,a_n$ [i]Proposed by Mohsen Jamali[/i]

Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.

Let $P $ be the intersection of $AC $ and $BD $ in isosceles trapezoid  $ABCD $ ($AB\parallel CD$ , $BC=AD $) . The circumcircle of triangle $ABP $ inersects $BC $ for the second time at $X $. Point $Y $ lies on $AX $ such that $DY\parallel BC $. Prove that $\hat {YDA} =2.\hat {YCA} $.

Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$, and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \neq A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ and $(c_1)$ be the circumcircles of the triangles $\triangle AEZ$ and $\triangle BEZ$, respectively. Let $(c_2)$ be an arbitrary circle passing through the points $A$ and $E$. Suppose $(c_1)$ meets the line $CZ$ again at the point $F$, and meets $(c_2)$ again at the point $N$. If $P$ is the other point of intersection of $(c_2)$ with $AF$, prove that the points $N$, $B$, $P$ are collinear.

$x,y,z \in \mathbb{R}^+$ and $x^5+y^5+z^5=xy+yz+zx$. Prove that $$3 \ge x^2y+y^2z+z^2x$$

Find all triplets natural numbers $(a, b, c)$ satisfied \[GCD(a, b) + LCM(a,b) = 2021^c\] with $|a - b|$ and $(a+b)^2 + 4$ are both prime number

Let $0 \le a$, $b \le 1$ be real numbers. Prove the following inequality: \[\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.\] [i](41th Austrian Mathematical Olympiad, regional competition, problem 1)[/i]

Let $n$ be a positive integer. David has six $n\times n$ chessboards which he arranges in an $n\times n\times n$ cube. Two cells are "aligned" if they can be connected by a path of cells $a=c_1,\ c_2,\ \dots,\ c_m=b$ such that all consecutive cells in the path share a side, and the sides that the cell $c_i$ shares with its neighbors are on opposite sides of the square for $i=2,\ 3,\ \dots\ m-1$. Two towers attack each other if the cells they occupy are aligned. What is the maximum amount of towers he can place on the board such that no two towers attack each other?

Karina, Leticia and Milena paint glass bottles and sell them as decoration. they had $100$ bottles, and they decorated them in such a way that each bottle was painted by a single person. After the finished, they put all the bottles on a table. In an oversight one of them pushed the table, falling and breaking exactly $\frac18$ of the bottles that Karina painted, $\frac13$ of the bottles that Milena, painted and $\frac16$ of the bottles that Leticia painted. In total, $82$ painted bottles remained unbroken. Knowing that the number of broken bottles that Milena had painted is equal to the average of the amounts of broken bottles painted by Karina and Leticia, how many bottles did each of them paint?

Let $n$ be a positive integer divisible by $4$. We consider the permutations $(a_1, a_2,...,a_n)$ of $(1,2,..., n)$ having the following property: for each j we have $a_i + j = n + 1$ where $i = a_j$ . Prove that there are exactly $\frac{ (\frac12 n)!}{(\frac14 n)!}$ such permutations.

Let $ \left(a_{n} \right)_{n \equal{} 1}^{\infty }$ be a sequence on real numbers such that $ a{}_{n \plus{} 1} \equal{} a_{n} a_{n \plus{} 2}$ for every $ n\ge 1$. The number of elements in the set $ \left\{a_{n} : n\ge 1\right\}$ cannot be $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$

In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $l$ at the same point $A,$ but they may be on either side of $l$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$? $\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi$

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

The sequence $ (a_n)_{n\ge 1}$ is defined by $ a_1 \equal{} 3$, $ a_2 \equal{} 11$ and $ a_n \equal{} 4a_{n\minus{}1}\minus{}a_{n\minus{}2}$, for $ n \ge 3$. Prove that each term of this sequence is of the form $ a^2 \plus{} 2b^2$ for some natural numbers $ a$ and $ b$.

Prove that among ten consecutive natural numbers there is always at least one, and at most four, numbers that are not divisible by any of the numbers $ 2 $, $ 3 $, $ 5 $, $ 7 $.

Three numbers $N,n,r$ are such that the digits of $N,n,r$ taken together are formed by $1,2,3,4,5,6,7,8,9$ without repetition. If $N = n^2 - r$, find all possible combinations of $N,n,r$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a non-decreasing function such that $f(y) - f(x) < y - x$ for all real numbers $x$ and $y > x$. The sequence $u_1,u_2,\ldots$ of real numbers is such that $u_{n+2} = f(u_{n+1}) - f(u_n)$ for all $n\geq 1$. Prove that for any $\varepsilon > 0$ there exists a positive integer $N$ such that $|u_n| < \varepsilon$ for all $n\geq N$.

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?