Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.

There is a board with three rows and $2023$ columns. In the first row the numbers are written from $1$ to $2023$, ordered from least to greatest. The devil writes those same numbers in the boxes in the second row, but ordered to his choice. Then, in each box in the third row he writes the difference between the two numbers already written in his own column (the largest minus the smallest). For example, if the first two boxes of a column are the numbers $21$ and $198$, in the third box it is written $198-21 = 177$. Explain why, no matter how the devil completed the second row of the board, it will never happen that multiplying them $2023$ numbers in the third row the result is odd.

Each cell of a $10 \times 10$ board is painted red, blue or white, with exactly twenty of them red. No two adjacent cells are painted in the same colour. A domino consists of two adjacent cells, and it is said to be good if one cell is blue and the other is white. (a) Prove that it is always possible to cut out $30$ good dominoes from such a board. (b) Give an example of such a board from which it is possible to cut out $40$ good dominoes. (c) Give an example of such a board from which it is not possible to cut out more than $30$ good dominoes.

Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$

$3k$ points are marked on the circumference. They divide it onto $3k$ arcs. Some $k$ of them have length $1$, other $k$ of them have length $2$, the rest $k$ of them have length $3$. Prove that some two of the marked points are the ends of one diameter.

Let $S_1, S_2, \ldots, S_{100}$ be finite sets of integers whose intersection is not empty. For each non-empty $T \subseteq \{S_1, S_2, \ldots, S_{100}\},$ the size of the intersection of the sets in $T$ is a multiple of the number of sets in $T$. What is the least possible number of elements that are in at least $50$ sets? [i]Proposed by Rishabh Das[/i]

$a_1, a_2,...,a_{2006}$ is a permutation of $1,2,...,2006$. Prove that $\prod_{i = 1}^{2006} (a_{i}^2-i) $ is a multiple of $3$. ($0$ is counted as a multiple of $3$)

Each of $100$ stones has a sticker showing its true weight. No two stones weight the same. Mischievous Greg wants to rearrange stickers so that the sum of the numbers on the stickers for any group containing from $1$ to $99$ stones is different from the true weight of this group. Is it always possible?

The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

There are $n$ fleas on an infinite sheet of triangulated paper. Initially the fleas are in different small triangles, all of which are inside some equilateral triangle consisting of $n^2$ small triangles. Once a second each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. For which natural numbers $n$ does there exist an initial configuration such that after a finite number of jumps all the $n$ fleas can meet in a single small triangle?

In the reality show [i]Big Sister Brasil[/i], it is said that there is a [i]treta[/i] if two people are friends with each other and enemies with a third one. For audience purposes, the broadcaster wants a lot of [i]tretas[/i]. If friendship and enmity are reciprocal relationships, given $n$ people, what is the maximum number of [i]tretas[/i]?

There are $n\ge 3$ particles on a circle situated at the vertices of a regular $n$-gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before. Let $s$ be the smallest number of collisions after which all particles return to their original positions. Find $s$. [i]Proposed by N.V. Tejaswi[/i]

What is the number of ways in which one can color the squares of a $4\times 4$ chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares?

A subset of $ n\times n$ table is called even if it contains even elements of each row and each column. Find the minimum $ k$ such that each subset of this table with $ k$ elements contains an even subset

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Recall that a plane divides $\mathbb{R}^3$ into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube $ABCD-A_1B_1C_1D_1$ lie on, and the four planes that the tetrahedron $ACB_1D_1$ lie on. How many regions do these ten planes split the space into?

In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?

Consider $n$ lines in the plane in general position, that is, there are not three of the $n$ lines that pass through the same point. Determine if it is possible to label the $k$ points where these lines are inserted with the numbers $1$ through $k$ (using each number exactly once), so that on each line, the labels of the $n-1$ points of that line are arranged in increasing order (in one of the two directions in which they can be traversed).

Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ). Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.

If five squares of a $3 \times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?

In a certain language there are only two letters, $A$ and $B$. We know that (i) There are no words of length $1$, and the only words of length $2$ are $AB$ and $BB$. (ii) A segment of length $n > 2$ is a word if and only if it can be obtained from a word of length less than $n$ by replacing each letter $B$ by some (not necessarily the same) word. Prove that the number of words of length $n$ is equal to $\frac{2^n +2\cdot (-1)^n}{3}$

Each of the numbers $1, 2, 3,... , 2006^2$ is placed at random into a cell of a $2006\times 2006$ board. Prove that there exist two cells which share a common side or a common vertex such that the sum of the numbers in them is divisible by $4$. (4)

Given $101$ distinct non-negative integers less than $5050$ show that one can choose four $a, b, c, d$ such that $a + b - c - d$ is a multiple of $5050$

A chess king tours an entire $8\times 8$ chess board, visiting each square exactly once and returning at last to his starting position. Prove that he made an even number of diagonal moves. (V Proizvolov)