A partition of a positive integer $n$ is a summing $n_1+\ldots+n_k=n$, where $n_1\ge n_2\ge\ldots\ge n_k$. Call a partition [i]perfect[/i] if every $m\le n$ can be represented uniquely as a sum of some subset of the $n_i$'s. How many perfect partitions are there of $n=307$?

For any natural $n$, $f(n)$ is the number of labeled digraphs with $n$ vertices such that for any vertex the number if in-edges is equal to the number of out-edges and the total of (in+out) edges is even. Let $g(n)$ be the odd-analogous of $f(n)$. Find $g(n)-f(n)$ with proof . [hide=original formulation] Dado $n$ natural, seja $f(n)$ o número de grafos rotulados direcionados com $n$ vértices de modo que em cada vértice o número de arestas que chegam é igual ao número de arestas que saem e o número de arestas total do grafo é par . Defina $g(n)$ analogamente trocando "par" por "ímpar" na definição acima. Calcule $f(n) - g (n)$. (Observação: Um grafo rotulado direcionado é um par $G = (V, E)$ onde $V = \{1, 2, …, n\}$ e $E$ é um subconjunto de $V^2 -\{(i, i); 0 < i < n + 1\}$).[/hide]

Boy A and $2016$ flags are on a circumference whose length is $1$ of a circle. He wants to get all flags by moving on the circumference. He can get all flags by moving distance $l$ regardless of the positions of boy A and flags. Find the possible minimum value as $l$ like this. Note that boy A doesn’t have to return to the starting point to leave gotten flags.

In one criminal kingdom, an underdeveloped state, the King decided to start a fight against corruption and, as an example, punish one of his $199$ ministers. The ministers were summoned to the palace and seated at a large round table. At first they wanted to find the one who had the most money in his bank account and declare him the main corrupt official. It takes $20$ minutes to determine the amount of money in the bank account of one minister. But the King ordered that the accused be found within four hours while he underwent medical procedures. According to the Noble Court Administrator, any minister can be accused, you just need to find a legal justification.The Chief Lawyer proposed that the first minister discovered, who has more money in his bank account than each of his two neighbors (one on the right and one on the left), be declared corrupt. How can one be sure to find a minister who meets this condition within the allotted $4$ hours? (During this time, it is possible to consistently determine the size of the bank accounts of no more than $12$ ministers. It is assumed that the amount of money in bank accounts is different.)

Let $S = \{A_1,A_2,\ldots ,A_k\}$ be a collection of subsets of an $n$-element set $A$. If for any two elements $x, y \in A$ there is a subset $A_i \in S$ containing exactly one of the two elements $x$, $y$, prove that $2^k\geq n$. [i]Yugoslavia[/i]

[b]a)[/b] Determine if it's possible write $6$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers. [b]b)[/b] Determine if it's possible write $8$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers.

A number of signal lights are equally spaced along a one-way railroad track, labeled in oder $ 1,2, \ldots, N, N \geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of $ K$ freight trains must be driven from Signal 1 to Signal $ N.$ Each train travels at a distinct but constant spped at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal $ N.$

For any positive integer $n$, let $f(n)$ be the number of possible choices of signs $+\ \text{or}\ - $ in the algebraic expression $\pm 1\pm 2\ldots \pm n$, such that the obtained sum is zero. Show that $f(n)$ satisfies the following conditions: a) $f(n)=0$ for $n=1\pmod{4}$ or $n=2\pmod{4}$. b) $2^{\frac{n}{2}-1}\le f(n)\le 2^n-2^{\lfloor\frac{n}{2}\rfloor+1}$, for $n=0\pmod{4}$ or $n=3\pmod{4}$. [i]Ioan Tomsecu[/i]

There are $k$ points in the interior of a pentagon. Together with the vertices of the pentagon they form a $(k + 5)$-element set $M$. The area of the pentagon is defined by connecting lines between the points of $M$ into sub-areas in such a way that it is divided into sub-areas in such a way that no sub-areas have a point on their interior of $M$ and contains exactly three points of $M$ on the boundary of each part. None of the connecting lines has a point in common with any other connecting line or pentagon side, which does not belong to $M$. With such a decomposition of the pentagon, there can be an even number of connecting lines (including the pentagon sides) go out? The answer has to be justified.

Michael and Steven are playing the card game War with a deck of $4$ cards numbered $1$ through $4.$ The deck is shuffled randomly and Michael gets a stack of $1$ card and Steven gets a stack of $3$ cards. In each round, the players reveal the top card from their stack, and the player whose card was higher collects both cards to the bottom of their stack in random order. A player wins when they get all four cards in their hand. The probability that Michael wins after exactly $5$ rounds is $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m + n.$

Mykhailo drew a triangular grid with side \( n \) for \( n \geq 2 \). It is formed from an equilateral triangle \( T \) with side length \( n \), by dividing each side into \( n \) equal parts. Then lines are drawn parallel to the sides of triangle \( T \), dividing it into \( n^2 \) equilateral triangles with side length \( 1 \), which we will call \textbf{cells}. Next, Oleksii writes some positive integer into each cell. Mykhailo receives 1 candy for each cell, where the number written is equal to the sum of all the numbers in the adjacent cells. Oleksii wants to arrange the numbers in such a way that Mykhailo receives the maximum number of candies. How many candies can Mykhailo receive under such conditions? In the figure below, an example is shown for \( n = 4 \) with 16 cells and numbers written inside them. For the numbers arranged as in the figure, Mykhailo receives 5 candies for the numbers \( 2 \) (the topmost cell), \( 8 \), \( 13 \), \( 12 \), and \( 11 \). [img]https://i.ibb.co/LrLks9q/Kyiv-MO-2025-R2-7-1.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

Find all positive integers $n \geq 3$ for which there exists a set $S$ which consists of rational numbers such that the following two conditions hold: 1) any rational number can be represented as the sum of at most $n$ elements of $S$ 2) there exists a rational number, which can not be represented as the sum of at most $n-1$ elements of $S$ (in the sum some elements can repeat) [i]M. Shutro, M. Zorka[/i]

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then a)$800000000$; b)$880000000$ by such replacements?

The $27$ points $(a,b,c)$ of the space are marked such that $a$, $b$ and $c$ take the values $0$, $1$ or $2$. We will call these points "junctures". Using $54$ rods of length $1$, all the joints that are at a distance of $1$ are joined together. A cubic structure of $2\times 2\times 2$ is thus formed. An ant starts from a juncture $A$ and moves along the rods; When it reaches a juncture it turns $90^\circ$ and changes rod. If the ant returns to $A$ and has not visited any juncture more than once except $A$, which it visited $2$ times, at the beginning of the walk and at the end of it, what is the greatest length that the path of the ant can have?

Let $m,n$ be integers so that $m \ge n > 1$. Let $F_1,...,F_k$ be a collection of $n$-element subsets of $\{1,...,m\}$ so that $F_i\cap F_j$ contains at most $1$ element, $1 \le i < j \le k$. Show that $k\le \frac{m(m-1)}{n(n-1)} $

We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.

The $n$ participants of a tournament are numbered with $0$ through $n - 1$. At the end of the tournament it turned out that for every team, numbered with $s$ and having $t$ points, there are exactly $t$ teams having $s$ points each. Determine all possibilities for the final score list.

Exactly 1600 Coconuts are distributed on exactly 100 monkeys, where some monkeys also can have 0 coconuts. Prove that, no matter how you distribute the coconuts, at least 4 monkeys will always have the same amount of coconuts. (The original problem is written in German. So, I apologize when I've changed the original problem or something has become unclear while translating.)

A number of unit discs are given inside a square of side $100$ such that (i) no two of the discs have a common interior point, and (ii) every segment of length $10$, lying entirely within the square, meets at least one disc. Prove that there are at least $400$ discs in the square.

Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns. Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.

Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.

The vertices of a regular $2n$-gon ($n \ge 3$) are labelled with the numbers $1,2,...,2n$ so that the sum of the numbers at any two adjacent vertices equals the sum of the numbers at the vertices diametrically opposite to them. Show that this is only possible if $n$ is odd.

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

On Babba Island they use a two-letter alphabet, $a$ and $b$, and every (finite) sequence of letters is a word. For each set $P$ of six words of $4$ letters each, we denote $N_P$ to the set of all words that do not contain any of the words of $P$ as a syllable (subword). Prove that if $N_P$ is finite, then all its words are of length less than or equal to $10$, and find a set $P$ such that $N_P$ is finite and contains at least one word of length $10$.

On the coast of a circular island there are eight different cities. Initially there are no routes between the cities. We have to construct five straight two-way routes, which do not intersect, so that from each city there are one or two routes. In how many ways can this happen?