Prove that the transformation product of the symmetry of center $(0, 0)$ with the symmetry of the axis, with the line of equation $x = y + 1$, can be expressed as a product of an axis symmetry the line $e$ by a translation of vector $\overrightarrow{v}$, with $e$ parallel to $\overrightarrow{v}$, . Determine a line $e$ and a vector $\overrightarrow{v}$, that meet the indicated conditions. have to be unique $e$ and $\overrightarrow{v}$,?

Solve the equation $mn =$ (gcd($m,n$))$^2$ + lcm($m, n$) in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.

Let $ABC$ be a triangle and $P$ be the midpoint of arc $BAC$ of circumcircle of triangle $ABC$ with orthocenter $H$. Let $Q, S$ be points such that $HAPQ$ and $SACQ$ are parallelograms. Let $T$ be the midpoint of $AQ$, and $R$ be the intersection point of the lines $SQ$ and $PB$. Prove that $AB$, $SH$ and $TR$ are concurrent. [i]Proposed by Dominik Burek - Poland[/i]

The Yamaimo family is moving to a new house, so they’ve packed their belongings into boxes, which weigh $100\text{ kg}$ in total. Mr. Yamaimo realizes that $99\%$ of the weight of the boxes is due to books. Later, the family unpacks some of the books (and nothing else). Mr. Yamaimo notices that now only $95\%$ of the weight of the boxes is due to books. How much do the boxes weigh now in kilograms?

Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$. [i] Proposed by Ilija Jovčevski, North Macedonia[/i]

It is allowed to perform the following transformations in the plane with any integers $a$: (1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$, (2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$. Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to: a) Vertices of a square, b) Vertices of a rectangle with unequal side lengths?

Let circles $\Omega$ and $\omega$ touch internally at point $A$. A chord $BC$ of $\Omega$ touches $\omega$ at point $K$. Let $O$ be the center of $\omega$. Prove that the circle $BOC$ bisects segment $AK$.

Suppose we have distinct positive integers $a, b, c, d$, and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence \begin{align*} ca &- db \\ ca^2 &- db^2 \\ ca^3 &- db^3 \\ ca^4 &- db^4 \\ &\vdots \end{align*} and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.

Call a set $S \subseteq \{0,1,\dots,14\}$ $\textit{sparse}$ if $x+1 \pmod{15}$ is not in $S$ whenever $x \in S$. Find the number of sparse sets $T$ such that the sum of the elements of $T$ is a multiple of 15.

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

Anna and Basilis play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$. Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

Find all natural numbers such that \[ n\sigma(n)\equiv 2\pmod {\phi( n)}\]

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.

Let $m$ and $n$ be integers greater than $1$ and $a_1 ,a_2 ,\ldots, a_{m+1}$ be real numbers. Prove that there exist real $n\times n$ matrices $A_1 ,A_2,\ldots, A_m$ such that (i) $\det(A_j) =a_j$ for $j=1,2,\ldots,m$ and (ii) $\det(A_1 +A_2 +\ldots+A_m)=a_{m+1}.$

show that the polynomial $x^4+3x^3+6x^2+9x+12$ cannot be written as the product of 2 polynomials of degree 2 with integer coefficients.

Inside triangle $ABC$ consider random point $O$. Prove that: $$E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O}$$ where $E_A,E_B,E_C$ the areas of triangle $BOC, COB, AOB$ respectively

In triangle $ABC$, the points $D$, $E$, and $F$ are the feet of the perpendiculars dropped from $A$, $B$, and $C$, respectively, onto the opposite sides. The point $X_A$ is such that a circle passing through $E$ and $F$ is tangent to the circumcircle of triangle $ABC$ at $X_A$, and $X_A$ is on a different side of $EF$ as $A$. Similarly, $X_B$ and $X_C$ are defined. Prove that the lines $AX_A$, $BX_B$, and $CX_C$ are concurrent.

Let $a,b,c$ be positive reals. Prove that $$a^ab^bc^c \geq a^bb^cc^a$$

Prove that the inequality $$\left(a_1+\cdots+a_n\right)\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\ge n^2$$ holds for any positive numbers $a_1,\ldots,a_n$ and determine when equality occurs.

There are $m$ boys and $n$ girls participate in a duet singing contest ($m,n\geq 2$). At the contest, each section there will be one show. And a show including some boy-girl duets where each boy-girl couple will sing with each other no more than one song and each participant will sing at least one song. Two show $A$ and $B$ are considered different if there is a boy-girl couple sing at show $A$ but not show $B$. The contest will end if and only if every possible shows are performed, and each show is performed exactly one time. a) A show is called [i]depend[/i] on a participant $X$ if we cancel all duets that $X$ perform, then there will be at least one another participant will not sing any song in that show. Prove that among every shows that depend on $X$, the number of shows with odd number of songs equal to the number of shows with even number of songs. b) Prove that the organizers can arrange the shows in order to make sure that the numbers of songs in two consecutive shows have different parity.

Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.

An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?

In an acute triangle $ABC$, the bisector $AL$, the altitude $BH$, and the perpendicular bisector of the side $AB$ intersect at one point. Find the value of the angle $BAC$.