Find the smallest positive integer $n$ having the property that for any $n$ distinct integers $a_1, a_2, \dots , a_n$ the product of all differences $a_i-a_j$ $(i < j)$ is divisible by $1991$.

For the Dürer final results announcement, four loudspeakers are used to provide sound in the hall. However, there are only two sockets in the wall from which the power comes. To solve the problem, Ádám got two extension cords and two power strips. One plug can be plugged into an extension cord, and two plugs can be plugged into a power strip. Gábor, in his haste before the announcement of the results, quickly plugs the $8$ plugs into the $8$ holes. Every possible way of plugging has the same probability, and it is also possible for Gábor to plug something into itself. What is the probability that all $4$ speakers will have sound at the results announcement? For the solution, give the sum of the numerator and the denominator in the simplified form of the probability. A speaker sounds when it is plugged directly or indirectly into the wall.

3. Let n > 1 be an integer and $a_1, a_2, . . . , a_n$ be positive reals with sum 1. a) Show that there exists a constant c ≥ 1/2 so that $\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c$, where $a_0 = 0$. b) Show that ’the best’ value of c is at least $\frac{\pi}{4}$.

Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$. [list=a] [*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$. [*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$. [/list]

$A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$?

a)Prove that any matrix $A\in \mathcal{M}_4(\mathbb{C})$ can be written as a sum of four matrices $B_1,B_2,B_3,B_4\in \mathcal{M}_4(\mathbb{C})$ with the rank equal to $1$. b)$I_4$ can't be written as a sum of less than four matrices with the rank equal to $1$. [i]Manuela Prajea & Ion Savu[/i]

Consider a square ABCD. A point P was selected on its diagonal AC. Let H be the orthocenter of the triangle APD, let M be the midpoint of AD and N be the midpoint of CD. Prove that PN is orthogonal to MH.

Line segment $\overline{AB}$ has perpendicular bisector $\overline{CD}$, where $C$ is the midpoint of $\overline{AB}$. The segments have lengths $AB = 72$ and $CD = 60$. Let $R$ be the set of points $P$ that are midpoints of line segments $\overline{XY}$ , where $X$ lies on $\overline{AB}$ and $Y$ lies on $\overline{CD}$. Find the area of the region $R$.

Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\ldots \ 2p\}$ are there, the sum of whose elements is divisible by $p$?

Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.

Let $n$ be a positive integer with $k$ digits. A number $m$ is called an $alero$ of $n$ if there exist distinct digits $a_1$, $a_2$, $\dotsb$, $a_k$, all different from each other and from zero, such that $m$ is obtained by adding the digit $a_i$ to the $i$-th digit of $n$, and no sum exceeds 9. For example, if $n$ $=$ $2024$ and we choose $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $3$, then $m$ $=$ $4177$ is an alero of $n$, but if we choose the digits $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $6$, then we don't obtain an alero of $n$, because $4$ $+$ $6$ exceeds $9$. Find the smallest $n$ which is a multiple of $2024$ that has an alero which is also a multiple of $2024$.

suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that $\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$ $\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$ $\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi

On the sides $AD$ and $BC$ of a rectangle $ABCD$ select points $M, N$ and $P, Q$ respectively such that $AM = MN = ND = BP = PQ = QC$. On segment $QC$ selected point $X$, different from the ends of the segment. Prove that the perimeter of $\vartriangle ANX$ is more than the perimeter of $\vartriangle MDX$.

Fill each cell with an integer from $1$-$7$ so each number appears exactly once in each row and column. In each ``cage" of three cells, the three numbers must be valid lengths for the sides of a non-degenerate triangle. Additionally, if a cage has an ``A", the triangle must be acute, and if the cage has an ``R", the triangle must be right. [asy] for(int i = 0; i < 8; ++i){ draw((0,i) -- (7,i)^^(i,0)--(i,7), gray(0.7)); } draw((2.1,6.1) -- (4.9, 6.1)--(4.9, 6.9) -- (2.1,6.9)--cycle); draw((5.1,6.1) -- (6.1, 6.1) -- (6.1, 5.1) -- (6.9, 5.1) -- (6.9, 6.9) --(5.1, 6.9) -- cycle); label(scale(0.5)*"R", (5.1, 6.9), SE); draw((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle); draw((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle); draw(shift((3,0))*((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle)); draw(shift((3,-1))*((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle)); label(scale(0.5)*"A", (6.1, 4.9), SE); draw(shift((2,-2))*((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle)); draw((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle); label(scale(0.5)*"R", (4.1, 3.9), SE); draw((0.1, 2.1) -- (0.1, 3.9) -- (1.9, 3.9) -- (1.9, 3.1) -- (0.9, 3.1) -- (0.9, 2.1) -- cycle); draw(shift((0, -3))*((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle)); label(scale(0.5)*"R", (1.1, 2.9), SE); draw(shift((-2, -6)) * ((2.1,6.1) -- (4.9, 6.1)--(4.9, 6.9) -- (2.1,6.9)--cycle)); label(scale(0.5)*"A", (0.1, 0.9), SE); draw(shift((0,-2))*((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle)); label(scale(0.5)*"A", (4.1, 1.9), SE); draw(shift((2,-2))*((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle)); [/asy]

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]