Find all functions \( f: \mathbb{R}^{+} \to \mathbb{R}^{+} \) such that: \[ f(x^2 + y) = xf(x) + \frac{f(y^2)}{y} \] for any positive real numbers \( x \) and \( y \).

Let \( n \geq 3 \) be an integer. For every two adjacent vertices \( A \) and \( B \) of a convex \( n \)-gon, we find a vertex \( C \) such that the angle \( \angle ACB \) is the largest, and write down the measure in degrees. Find the smallest possible value of the sum of the written \( n \) numbers.

The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \). a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \). b) Does there exist a non-constant bounded sequence with the above properties? (A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)

Let \( n \geq 4 \) be a positive integer. Initially, each of \( n \) girls knows one piece of gossip that no one else knows, and they want to share them. For greater security, to avoid being spied, they only talk in pairs, and in a conversation, each girl shares all the gossip she knows so far with the other one. What is the minimum number of conversations needed so that every girl knows all the gossip?

The positive integers \( a \) and \( b \) are coprime and such that there exist positive integers \( m_2 \) and \( m_5 \) for which \( am_2 + b \) is a perfect square of a positive integer, and \( am_5 + b \) is a perfect fifth power of a positive integer. Does there always exist a positive integer \( n \) for which \( an + b \) is a perfect \( k \)-th power of a positive integer, if: a) \( k = 7 \); b) \( k = 10 \)?

A triangle \( ABC \) is given with centers \( O \) and \( I \) of the circumscribed and inscribed circles, respectively. Point \( A_1 \) is the reflection of \( A \) with respect to \( I \). Point \( A_2 \) is such that lines \( BA_1 \) and \( BA_2 \) are symmetric with respect to \( BI \), and lines \( CA_1 \) and \( CA_2 \) are symmetric with respect to \( CI \). Prove that \( AO^2 = |A_2O^2 - A_2I^2| \).

The Young Scientist and the Old Scientist play the following game, taking turns in an alternating fashion, with the Young Scientist starting first. The player on turn fills in one of the stars in the equation \[ x^4 + *x^3 + *x^2 + *x + * = 0 \] with a positive real number. Who has a winning strategy if the goals of the players are: a) the Young Scientist - to make the resulting equation have no real roots, and the Old Scientist -- to make it have real roots? b) the Young Scientist - to make the resulting equation have real roots, and the Old Scientist -- to make it have none?

Each cell in a \( 2024 \times 2024 \) table contains the letter \( A \) or \( B \), with the number of \( A \)'s in each row being the same and the number of \( B \)'s in each column being the same. Alexandra and Boris play the following game, alternating turns, with Alexandra going first. On each turn, the player chooses a row or column and erases all the letters in it that have not yet been erased, as long as at least one letter is erased during the turn, and at the end of the turn, at least one letter remains in the table. The game ends when exactly one letter remains in the table. Alexandra wins the game if the letter is \( A \), and Boris wins if it is \( B \). What is the number of initial tables for which Alexandra has a winning strategy?

Given a prime number \( p \geq 3 \) and a positive integer \( m \), find the smallest positive integer \( n \) with the following property: for every positive integer \( a \), which is not divisible by \( p \), the sum of the natural divisors of \( a^n \) greater than 1 is divisible by \( p^m \).

For arbitrary real numbers \( x_1,x_2,\ldots,x_n \), prove that \[ \left(\max_{1\leq i \leq n}x_i \right)^2 + 4\sum_{i=1}^{n-1}\left(\max_{1\leq j \leq i}x_j\right)\left(x_{i+1}-x_i\right) \leq 4x_n^2. \]

Let \( X \) be an arbitrary point on the side \( BC \) of triangle \( ABC \). The point \( M \) on the ray \( AB^\to \) beyond \( B \), the point \( N \) on the ray \( AC^\to \) beyond \( C \), and the point \( K \) inside \( ABC \) are such that \( \angle BMX = \angle CNX = \angle KBC = \angle KCB \). The line through \( A \), parallel to \( BC \), intersects the line \( KX \) at point \( P \). Prove that the points \( A \), \( P \), \( M \), \( N \) lie on a circle.

A collection of \( n \) balls is distributed in several boxes, with no box containing 100 or more balls. In one move, we can remove several (at least one, possibly all) balls from one box. Find the smallest positive integer \( n \) with the following property: regardless of the distribution, we can make moves such that each non-empty box contains the same number of balls and the total number of remaining balls is at least 100.

Depending on the real number \( a \), find all polynomials \( P(x) \) with real coefficients such that \[ (x^3 - ax^2 + 1)P(x) = (x^3 + ax^2 + 1)P(x-1) \]for every real number \( x \).

Each of 9 girls participates in several (one or more) theater groups, so that there are no two identical groups. Each of them is randomly assigned a positive integer between 1 and 30 inclusive. We call a group \textit{small} if the sum of the numbers of its members does not exceed the sum of any other group. Prove that regardless of which girl participates in which group, the probability that after receiving the numbers there will be a unique small group is at least \( \frac{7}{10} \).

A set \( S \) of two or more positive integers is called almost closed under addition if the sum of any two distinct elements of \( S \) also belongs to \( S \). Let \( P(x) \) be a polynomial with integer coefficients for which there exists an almost closed under addition set \( S \), such that for any two distinct \( a \) and \( b \) from \( S \), the numbers \( P(a) \) and \( P(b) \) are coprime. Prove that \( P \) is a constant.

Let \( ABC \) and \( A_1B_1C_1 \) be two triangles such that the segments \( AA_1 \) and \( BC \) intersect, the segments \( BB_1 \) and \( AC \) intersect, and the segments \( CC_1 \) and \( AB \) intersect. If it is known that there exists a point \( X \) inside both triangles such that \[ \begin{aligned} \angle XAB &= \angle XA_1B_1, &\angle XBC &= \angle XC_1A_1, &\angle XCA &= \angle XB_1C_1,\\ \angle XAC &= \angle XB_1A_1, &\angle XBA &= \angle XA_1C_1, &\angle XCB &= \angle XC_1B_1. \end{aligned} \]Prove that the lines \( AC_1 \), \( BB_1 \), and \( CA_1 \) are concurrent or parallel.

Find all quadruples \((a,b,c,d)\) of positive integers such that \(\displaystyle \frac{ac+bd}{a+c}\) and \(\displaystyle \frac{bc-ad}{b-d}\) are equal to the prime number \(90121\).

Let \(m,n\) and \(a\) be positive integers. Lumis has \(m\) cards, each with the number \(n\) written on it, and an infinite number of cards with each of the symbols addition, subtraction, multiplication, division, opening, and closing brackets. Umbra has composed an arithmetic expression with them, whose value is a positive integer less than \(\displaystyle\frac{n}{2^m}\). Prove that if \(n\) is replaced everywhere by \(a\), then the resulting expression will have the same value as before or will be undefined due to division by zero.

Given a parallelogram \(ABCD\). Let \(\ell_1\) be the line through \(D\), parallel to \(AC\), and \(\ell_2\) the external bisector of \(\angle ACD\). The lines \(\ell_1\) and \(\ell_2\) intersect at \(E\). The lines \(\ell_1\) and \(AB\) intersect at \(F\), and the line \(\ell_2\) intersects the internal bisector of \(\angle BAC\) at \(X\). The line \(BX\) intersects the circumcircle of triangle \(EFX\) at a second point \(Y\). The internal bisector of \(\angle ACD\) intersects the circumcircle of triangle \(ACX\) at a second point \(Z\). Prove that the quadrilateral \(DXYZ\) is inscribed in a circle.

At the wedding of two Bulgarian nationals in mathematics, every guest who gave a positive integer \(n\), not yet given by another guest, which divides \(3^n-3\) but does not divide \(2^n-2\), received a prize. If there were an infinite number of guests, would the newlyweds theoretically need an infinite number of gifts?

Find all functions \(f:\mathbb{R}^{+} \to \mathbb{R}^{+}\) such that \[ f(x) > x \ \ \text{and} \ \ f(x-y+xy+f(y)) = f(x+y) + xf(y) \]for arbitrary positive real numbers \(x\) and \(y\).

Prove that for some positive integer \(N\), \(N\) points can be chosen on a circle such that there are at least \(1000N^2\) unordered quadruples \((A,B,C,D)\) of distinct selected points for which \(\displaystyle \frac{AC}{BC} = \frac{AD}{BD}\).

The positive integers from 1 to \(n\) are arranged in a sequence, initially in ascending order. In one move, we can swap the positions of two of the numbers, provided they share a common divisor greater than 1. Let \(s_n\) be the number of sequences that can be obtained with a finite number of moves. Prove that \(s_n = a_n!\), where the sequence of positive integers \((a_n)_{n\geq 1}\) is such that for any \(\delta > 0\), there exists an integer \(N\), for which for all \(n\geq N\), the following is true: \[ n - \left(\frac{1}{2}+\delta\right)\frac{n}{\log n} < a_n < n - \left(\frac{1}{2}-\delta\right)\frac{n}{\log n}. \]

Three piles of stones are given, initially containing 2000, 4000, and 4899 stones respectively. Ali and Baba play the following game, taking turns, with Ali starting first. In one move, a player can choose two piles and transfer some stones from one pile to the other, provided that at the end of the move, the pile from which the stones are moved has no fewer stones than the pile to which the stones are moved. The player who cannot make a move loses. Does either player have a winning strategy, and if so, who?

Does there exist a polynomial \( P(x,y) \) in two variables with real coefficients, such that the following two conditions hold: 1) \( P(x,y) = P(x, x-y) = P(y-x, y) \) for any real numbers \( x \) and \( y \); 2) There does not exist a polynomial \( Q(z) \) in one variable with real coefficients such that \( P(x,y) = Q(x^2 - xy + y^2) \) for any real numbers \( x \) and \( y \)?

Given an acute-angled triangle $ABC$ ($AB \neq AC$) with orthocenter $H$, circumcenter $O$, and midpoint $M$ of side $BC$. The line $AM$ intersects the circumcircle of triangle $BHC$ at point $K$, with $M$ between $A$ and $K$. The segments $HK$ and $BC$ intersect at point $N$. If $\angle BAM = \angle CAN$, prove that the lines $AN$ and $OH$ are perpendicular.

Let $(a_n)_{n\geq 1}$ be a (not necessarily strictly) increasing sequence of positive integers, such that $a_n \leq 1000n^{0.999}$ for every positive integer $n$. Prove that there exist infinitely many positive integers $n$ for which $a_n$ divides $n$.

Let $m > n$ be positive integers. In the host country of the International Olympiad in Informatics this year, there are coins of $1$, $2$, $\ldots$, $n$ alexandrias, lira banknotes, each worth $m$ alexandrias, and pharaoh banknotes, each worth $m+n$ alexandrias. Let $A$ be a positive integer. Boris wants to exchange the amount $A$ using coins and lira, using no more than $m-1$ coins, while Vesko wants to exchange the amount $A$ using coins and pharaohs, using no more than $m$ coins. Prove that regardless of the value of $A$, the number of ways for each of them to fulfill their desire is the same.

The function $f: A \rightarrow A$ is such that $f(x) \leq x^2 \mbox{ and } f(x+y) \leq f(x) + f(y) + 2xy$ for any $x, y \in A$. a) If $A = \mathbb{R}$, find all functions satisfying the conditions. b) If $A = \mathbb{R}^{-}$, prove that there are infinitely many functions satisfying the conditions. (With $\mathbb{R}^{-}$ we denote the set of negative real numbers.)

Let $P(x)$ be a polynomial in one variable with integer coefficients. Prove that the number of pairs $(m,n)$ of positive integers such that $2^n + P(n) = m!$, is finite.

Let $P$ be an arbitrary point on the incircle $k$ of triangle $ABC$ with center $I$, different from the points of tangency with its sides. The tangent to $k$ at $P$ intersects the lines $BC$, $AC$, $AB$ at points $A_0$, $B_0$, $C_0$, respectively. The lines through $A_0$, $B_0$, $C_0$, parallel to the bisectors of the angles $\angle BAC$, $\angle ABC$, $\angle ACB$, form a triangle $\Delta$. Prove that the line $PI$ is tangent to the circumcircle of $\Delta$.

Let $n \geq 2$ be a positive integer. In a mathematics competition, there are $n+1$ students, with one of them being a hacker. The competition is conducted as follows: each receives the same problem with an open-ended answer, has 5 minutes to give their own answer, after which all answers are submitted simultaneously, the correct answer is announced, then they receive a new problem, and so on. The hacker cheats by using spy cameras to see the answers of the other participants. A correct answer gives 1 point, while a wrong answer gives -1 point to everyone except the hacker; for him, it's 0 points because he managed to hack the scoring system. Prove that regardless of the total number of problems, if at some point the hacker is ahead of the second-place contestant by at least $2^{n-2} + 1$ points, then he has a strategy to ensure he will be the sole winner by the end of the competition.

Let \( n \geq 2 \) be a positive integer. Find all \( n \)-tuples \( (a_1, \ldots, a_n) \) of complex numbers such that the numbers \( a_1 - 2a_2 \), \( a_2 - 2a_3 \), $\ldots$ , \( a_{n-1} - 2a_n \), \( a_n - 2a_1 \) form a permutation of the numbers \( a_1, \ldots, a_n \).

Let \( p \neq 3 \) be a prime number. Prove that there exist natural numbers \( a \), \( b \), \( c \), \( d \), none of which are divisible by \( p \), such that \( a^2 + 3b^5 + 5c^6 + 7d^7 \) is divisible by \( p^{1000} \).

Find all functions \( f:\mathbb{Z} \to \mathbb{Z} \) such that \[ f(x + f(y) - 2y) + f(f(y)) = f(x) \]for all integers \( x \) and \( y \).

The diagonals \( AD \), \( BE \), and \( CF \) of a hexagon \( ABCDEF \) inscribed in a circle \( k \) intersect at a point \( P \), and the acute angle between any two of them is \( 60^\circ \). Let \( r_{AB} \) be the radius of the circle tangent to segments \( PA \) and \( PB \) and internally tangent to \( k \); the radii \( r_{BC} \), \( r_{CD} \), \( r_{DE} \), \( r_{EF} \), and \( r_{FA} \) are defined similarly. Prove that \[ r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}. \]

An infinite grid with two rows is divided into unit squares. One of the cells in the second row is colored red and all other cells in the grid are white. Initially, we are in the red cell. In one move, we can move from one cell to an adjacent cell (sharing a side). Find the number of sequences of \( n \) moves such that no cell is visited more than once. (In particular, it is not allowed to return to the red cell after several moves.)

The positive integers \( a \), \( b \), \( c \), \( d \) are such that \( (a+c)(b+d) = (ab-cd)^2 \). Prove that \( 4ad + 1 \) and \( 4bc + 1 \) are perfect squares of natural numbers.

Consider a finite undirected graph in which each edge belongs to at most three cycles. Prove that its vertices can be colored with three colors so that any two vertices connected by an edge have different colors. (A cycle in a graph is a sequence of distinct vertices \( v_1, v_2, \ldots, v_k \), \( k \geq 3 \), such that \( v_i v_{i+1} \) is an edge for each \( i = 1, 2, \ldots, k \); we consider \( v_{k+1} = v_1 \). The edges \( v_i v_{i+1} \) belong to the cycle.)

In space, there are \( 13 \) points, no four of which lie in the same plane. Three of the points are colored blue, and the triangle with these points as vertices will be called a blue triangle. The remaining \( 10 \) points are colored red. We say that a triangle with three red vertices is attached to the blue triangle if the boundary of the red triangle intersects the blue triangle (either in its interior or on its boundary) at exactly one point. Is it possible for the number of attached triangles to be \( 33 \)?