For all positive integers $n$ Prove that one can find pairwise coprime integers $a,b,c>n$ such that the set of prime divisors of the numbers $a+b+c$ and $ab+bc+ac$ coincides. Proposed by [i]Mohsen Jamali[/i] and [i]Hesam Rajabzadeh[/i]

Given a rectangle $ABCD$, determine two points $K$ and $L$ on the sides $BC$ and $CD$ such that the triangles $ABK, AKL$ and $ADL$ have same area.

Juana and Juan have to write each one an ordered list of fractions so that the two lists have the same number of fractions and that the difference between the sum of all the fractions from Juana's list and the sum of all fractions from Juan's list is greater than $123$. The fractions in Juana's list are $$\frac{1^2}{1}, \frac{2^2}{3},\frac{3^2}{5},\frac{4^2}{7},\frac{5^2}{9},...$$ And the fractions in John's list are $$\frac{1^2}{3}, \frac{2^2}{5},\frac{3^2}{7},\frac{4^2}{9},\frac{5^2}{11},...$$ Find the least amount of fractions that each one must write to achieve the objective.

Let $S$ be the set of monic polynomials in $x$ of degree 6 all of whose roots are members of the set $\{ -1, 0, 1\}$. Let $P$ be the sum of the polynomials in $S$. What is the coefficient of $x^4$ in $P(x)$?

Let $a,b,c\geq 0$ and $a^2+b^2+c^2\leq 1.$ Prove that$$\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1}+3abc<\sqrt 3$$

Find all positive integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{n-1}}{a_n}, \] where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$.

Circle $O$ is inscribed inside a non-isosceles trapezoid $JHMT$, tangent to all four of its sides. The longer of the two parallel sides of $JHMT$ is $\overline{JH}$ and has a length of $24$ units. Let $P$ be the point where $O$ is tangent to $\overline{JH}$, and let $Q$ be the point where $O$ is tangent to $\overline{MT}$. The circumcircle of $\vartriangle JQH$ intersects $O$ a second time at point $R$. $\overleftrightarrow{QR}$ intersects $\overleftrightarrow{JH}$ at point $S$, $35$ units away from $P$. The points inside $JHMT$ at which $\overline{JQ}$ and $\overline{HQ}$ intersect $O$ lie $\frac{63}{4}$ units apart. The area of $O$ can be expressed as $\frac{m\pi}{n}$ , where $\frac{m}{n}$ is a common fraction. Compute $m + n$.

[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes \[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\] [b](b)[/b] Find the rearrangement that minimizes $Q.$

The Sultan gathered $300$ court sages and offered them a test. There are caps of $25$ different colors, known in advance to the sages. The Sultan said that one of these caps will be put on each of the sages, and if for each color write the number of caps worn, then all numbers will be different. Every sage can see the caps of the other sages, but not own cap. Then all the sages will simultaneously announce the supposed color of their cap. Can sages advance agree to act in such a way that at least $150$ of them are guaranteed to name a color right?

A student divides all $30$ marbles into $5$ boxes numbered $1, 2, 3, 4, 5$ (after being divided, there may be a box with no marbles). a) How many ways are there to divide marbles into boxes (are two different ways if there is a box with a different number of marbles)? b) After dividing, the student paints those $30$ marbles by a number of colors (each with the same color, one color can be painted for many marbles), so that there are no $2$ marbles in the same box. have the same color and from any $2$ boxes it is impossible to choose $8$ marbles painted in $4$ colors. Prove that for every division, the student must use no less than $10$ colors to paint the marbles. c) Show a division so that with exactly $10$ colors the student can paint the marbles that satisfy the conditions in question b).

Let $AL$ be the inner bisector of triangle $ABC$. The circle centered at $B$ with radius $BL$ meets the ray $AL$ at points $L$ and $E$, and the circle centered at $C$ with radius $CL$ meets the ray $AL$ at points $L$ and $D$. Show that $AL^2 = AE\times AD$. [i](Proposed by Mykola Moroz)[/i]

How many integers $1\le x \le 2021$ make the value of the expression $$\frac{2x^3 - 6x^2 - 3x -20}{5(x - 4)}$$ an integer?

In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin \left( \angle DAE \right)$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$.

In quadrilateral $ ABCD$, it is given that $ \angle A \equal{} 120^\circ$, angles $ B$ and $ D$ are right angles, $ AB \equal{} 13$, and $ AD \equal{} 46$. Then $ AC \equal{}$ $ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 65 \qquad \textbf{(E)}\ 72$

Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Let be three natural numbers $ k,m,n $ an $ m\times n $ matrix $ A, $ an $ n\times m $ matrix $ B, $ and $ k $ complex numbers $ a_0,a_1,\ldots ,a_k $ such that the following conditions hold. $ \text{(i)}\quad m\ge n\ge 2 $ $ \text{(ii)}\quad a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m $ $ \text{(iii)}\quad a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n $ Prove that $ a_0=0. $

The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$? $\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$

Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[\left\lfloor\dfrac{2008}1\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^2}2\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^3}3\right\rfloor,\,\,\,\,\,\,\,\,\,\left\lfloor\dfrac{2008^4}4\right\rfloor,\,\,\,\,\,\,\,\,\,\ldots.\]

$x,y$ are naturals. $GCM(x^7,y^4)*GCM(x^8,y^5)=xy$ Prove that $xy$ is cube

Let $n$ be an odd positive integer such that both $\phi(n)$ and $\phi (n+1)$ are powers of two. Prove $n+1$ is power of two or $n=5$.

Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that \[ \frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|. \] When does equality hold?

There are real numbers $a,b$ and $c$ and a positive number $\lambda$ such that $f(x)=x^3+ax^2+bx+c$ has three real roots $x_1, x_2$ and $x_3$ satisfying $(1) x_2-x_1=\lambda$ $(2) x_3>\frac{1}{2}(x_1+x_2)$. Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$