Let m be a positive integer, and let $S=\{a_1=1, a_2, ..., a_m\}$ be a set of positive integers. Prove that there exists a positive integer $n$ with $n\le m$ and a set $T={b_1, b_2, ..., b_n}$ of positive integers such that (a) all the subsets of $T$ have distinct sums of elements; (b) each of the numbers $a_1$, $a_2$, ..., $a_m$ is the sum of the elements of a subset of $T$.

A pawn is put on each of $2n$ arbitrary selected cells of an $n\times n$ board ($n>1$). Prove that there are four cells that are marked with pawns and whose centers form a parallelogram.

A chocolate bar has five lengthwise dents and eight crosswise ones, which can be used to break up the bar into sections (one can get a total of $ 9 \times 6 = 54$ cells). Two players play the following game with such a bar. At each move (the two players move alternatively) one player breaks off a section of width one from the bar along a single dent and eats it, the other player does the same with what’s left of the bar, and so on. When one of the players breaks up a section of width two into two strips of width one, he eats one of the strips and the other player eats the other strip. Prove that the player who has the first move can play so as to eat at least $6$ cells more than his opponent (no matter how his opponent plays). (R Fedorov)

Consider a set of $16$ points arranged in $4 \times 4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

Ada rolls a standard $4$-sided die $5$ times. The probability that the die lands on at most two distinct sides can be written as $ \frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$

Set $S=\{1,2,...,n\}$ is firstly divided on $m$ disjoint nonempty subsets, and then on $m^2$ disjoint nonempty subsets. Prove that some $m$ elements of set $S$ were after first division in same set, and after the second division were in $m$ different sets

There are $2020$ positive integers written on a blackboard. Every minute, Zuming erases two of the numbers and replaces them by their sum, difference, product, or quotient. For example, if Zuming erases the numbers $6$ and $3$, he may replace them with one of the numbers in the set $\{6+3, 6-3, 3-6, 6\times 3, 6\div 3, 3\div 6\}$ $= \{9, 3, 3, 18, 2, \tfrac 12\}$. After $2019$ minutes, Zuming writes the single number $-2020$ on the blackboard. Show that it was possible for Zuming to have ended up with the single number $2020$ instead, using the same rules and starting with the same $2020$ integers. [i]Proposed by Zhuo Qun (Alex) Song[/i]

In two neigbouring cells(dimensions $1\times 1$) of square table $10\times 10$ there is hidden treasure. John needs to guess these cells. In one $\textit{move}$ he can choose some cell of the table and can get information whether there is treasure in it or not. Determine minimal number of $\textit{move}$'s, with properly strategy, that always allows John to find cells in which is treasure hidden.

For a family gathering, $8$ people order one dish each. The family sits around a circular table. Find the number of ways to place the dishes so that each person’s dish is either to the left, right, or directly in front of them. [i]Proposed by Nicole Sim[/i]

Show that we can find $\alpha>0$ such that, given any point $P$ inside a regular $2n+1$-gon which is inscribed in a circle radius $1$, we can find two vertices of the polygon whose distance from $P$ differ by less than $\frac1n-\frac\alpha{n^3}$.

On an $8\times 8$ chessboard, place a stick on each edge of each grid (on a common edge of two grid only one stick will be placed). What is the minimum number of sticks to be deleted so that the remaining sticks do not form any rectangle?

Let $G$ be a tournoment such that it's edges are colored either red or blue. Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.

There are $2024$ mathematicians sitting in a row next to the river Tisza. Each of them is working on exactly one research topic, and if two mathematicians are working on the same topic, everyone sitting between them is also working on it. Marvin is trying to figure out for each pair of mathematicians whether they are working on the same topic. He is allowed to ask each mathematician the following question: “How many of these 2024 mathematicians are working on your topic?” He asks the questions one by one, so he knows all previous answers before he asks the next one. Determine the smallest positive integer $k$ such that Marvin can always accomplish his goal with at most $k$ questions.

A ghost leg is a game with some vertical lines and some horizontal lines. A player starts at the top of the vertical line and go downwards, and always walkthrough a horizontal line if he encounters one. We define a layer is some horizontal line with the same height and has no duplicated endpoints. Find the smallest number of layers needed to grant that you can walk from $(1, 2, \ldots , n)$ on the top to any permutation $(\sigma_1, \sigma_2, \ldots, \sigma_n)$ on the bottom.

Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?

In a circumference there are $99$ natural numbers. If $a$ and $b$ are two consecutive numbers in the circle, then they must satisfies one of the following conditions: $a-b=1, a-b=2$ or $\frac{a}{b}=2$. Prove that, in the circle exists a number multiple of $3$.

A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?

Determine if there exists a triangle that can be cut into $101$ congruent triangles.

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?

At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw score and zero points for the defeat. At the end of the tournament it turned out that the sum of the winning points of all teams was $50$. (a) How many teams participated in this tournament? (b) How big is the difference between the team with the highest number and the number of points won?

Let \( G \) be a graph colored using \( k \) colors. We say that a vertex is [b]forced[/b] if it has neighbors in all the other \( k - 1 \) colors. Prove that for any \( 2024 \)-regular graph \( G \) that contains no triangles or quadrilaterals, there exists a coloring using \( 2025 \) colors such that at least \( 1013 \) of the colors have a forced vertex of that color. Note: The graph coloring must be valid, this means no \( 2 \) vertices of the same color may be adjacent.

In several cells of a square grid there live hedgehogs. Every hedgehog multiplies the number of hedgehogs in its row by the number of hedgehogs in its column. Is it possible that all the hedgehogs get distinct numbers?

Given a set $P$ of $n>100$ points on the plane such that no three of them are collinear, and a set $S$ of $20n$ distinct segments, each joining a pair of points from $P$. Prove that there exists a line not passing through a point from $P$ and intersecting at least $200$ segments from $S$.