Suppose $n$ houses are to be assigned to $n$ people. Each person ranks the houses in the order of preference, with no ties. After the assignment is made, it is observed that every other assignment would assign to at least one person a less preferred house. Prove that there is at least one person who received the house he/she preferred most under this assignment.

Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.

Prove that for each $m\geq1$: \[\sum_{|k|<\sqrt m}\binom{2m}{m+k}\geq 2^{2m-1}\] [hide="Hint"]Maybe probabilistic method works[/hide]

We color all vertexs of a convex polygon with $10$ vertexs by $2$ colors: red and blue $($each vertex is colored by $1$ color$).$ How many ways to color all the vertexs such that there are no $2$ adjacent vertex that are both colored red?

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

Given $k \in \mathbb{N}^+$. A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have (i) $168 \in I_i$; (ii) $\forall x, y \in I_i$, we have $x-y \in I_i$. Determine the number of $k-chain$ in total.

Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square $O$. A square is called [i]singular[/i] if $100$ is written in it and $101$ is written in all four squares sharing a side with it. How many singular squares are there?

Let $n$ be an even positive integer. We consider rectangles with integer side lengths $k$ and $k +1$, where $k$ is greater than $\frac{n}{2}$ and at most equal to $n$. Show that for all even positive integers $ n$ the sum of the areas of these rectangles equals $$\frac{n(n + 2)(7n + 4)}{24}.$$

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

4. A positive integer n is given.Find the smallest $k$ such that we can fill a $3*k$ gird with non-negative integers such that: $\newline$ $i$) Sum of the numbers in each column is $n$. $ii$) Each of the numbers $0,1,\dots,n$ appears at least once in each row.

[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.

Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number $x$ where $0 < x < 1$. In each move, Brian chooses a positive number $d$. Alice must move the checker to either $x + d$ or $x - d$. If it lands on $0$ or $1$, Brian wins. Otherwise the game proceeds to the next move. For which values of $x$ does Brian have a strategy which allows him to win the game in a finite number of moves?

There are $41$ letters on a circle, each letter is $A$ or $B$. It is allowed to replace $ABA$ by $B$ and conversely, as well as to replace $BAB$ by $A$ and conversely. Is it necessarily true that it is possible to obtain a circle containing a single letter repeating these operations? Maxim Didin

Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k \ ?$

Given a board $3\times3$ and $9$ cards with some numbers (known to the players). Two players, in turn, put those cards on the board. The first wins if the sum of the numbers in the first and the third row is greater than in the first and the third column. Prove that it doesn't matter what numbers are on the cards, but if the first plays the best way, the second can not win.

In each square of a $100 \times 100$ board there is written an integer. The allowed operation is to choose four squares that form the figure or any of its reflections or rotations, and add $1$ to each of the four numbers. The aim is, through operations allowed, achieving a board with the smallest possible number of different residues modulo $33$. What is the minimum number that can be achieved with certainty?

Let $n \geq 2$ be a positive integer and let $A, B$ be two finite sets of integers such that $|A| \leq n$. Let $C$ be a subset of the set $\{(a, b) | a \in A, b \in B\}$. Achilles writes on a board all possible distinct differences $a-b$ for $(a, b) \in C$ and suppose that their count is $d$. He writes on another board all triplets $(k, l, m)$, where $(k, l), (k, m) \in C$ and suppose that their count is $p$. Show that $np \geq d^2.$

Is it possible to disassemble and reassemble a $4\times 4\times 4$ Rubik's Cuble in at least $577,800$ non-equivalent ways? Notes: 1. When we reassemble the cube, a corner cube has to go to a corner cube, an edge cube must go to an edge cube and a central cube must go to a central cube. 2. Two positions of the cube are called equivalent if they can be obtained from one two another by rotating the faces or layers which are parallel to the faces.

Anis, Banu, and Cholis are going to play a game. They are given an $n\times n$ board consisting of $n^2$ unit squares, where $n$ is an integer and $n > 5$. In the beginning of the game, the number $n$ is written on each unit square. Then Anis, Banu, and Cholis take turns playing the game, repeatedly in that order, according to the following procedure: On every turn, an arrangement of $n$ squares on the same row or column is chosen, and every number from the chosen squares is subtracted by $1$. The turn cannot be done if it results in a negative number, that is, no arrangement of $n$ unit squares on the same column or row in which all of its unit squares contain a positive number can be found. The last person to get a turn wins. Determine which player will win the game.

A point $M (x, y)$ of the Cartesian plane of $xOy$ coordinates is called [i]lattice [/i] if it has integer coordinates. Each lattice point is colored red or blue. Prove that in the plan there is at least one rectangle with lattice vertices of the same color.

On an infinite checkered plane $100$ chips in form of a $10\times 10$ square are given. These chips are rearranged such that any two adjacent (by side) chips are again adjacent, moreover no two chips are in the same cell. Prove that the chips are again in form of a square.

A square has $2$ diagonals. A regular pentagon has $5$ diagonals. $n$ is the smallest positive integer such that a regular $n$-gon has greater than or equal to $2024$ diagonals. What is the sum of the digits of $n$? \[\mathrm a. ~10\qquad \mathrm b. ~11 \qquad \mathrm c. ~12 \qquad\mathrm d. ~13\qquad\mathrm e. ~14\]

Alina and Bogdan play a game on a $2\times n$ rectangular grid ($n\ge 2$) whose sides of length $2$ are glued together to form a cylinder. Alternating moves, each player cuts out a unit square of the grid. A player loses if his/her move causes the grid to lose circular connection (two unit squares that only touch at a corner are considered to be disconnected). Suppose Alina makes the first move. Which player has a winning strategy?

On a $2004\times 2004$ chessboard we place $2004$ white knights$^1$ in the upper row, and $2004$ black ones in the lowest row. After a finite number of regular chess moves$^2$ , we get the opposite situation where the black ones are on the top and the white ones on the bottom lines. In a [i]turn [/i] we make a move with each of the pieces of a color. If you know that each square except those on which the knights originally lie, must not be used more than once in this process, and that after each turn no $2$ knights of the same color can be attacking each other$^3$ , determine the number of ways in which this can be accomplished. $^1$ also known as horses $^2$ the knight can be moved either one square horizontally and two vertically or two squares horizontally and one vertically, in any direction on both horizontal and vertical lines $^3$ a knight is attacking another knight, if in one chess move, the first one can be placed on the second one’s place

Ivan has a $n \times n$ board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let $d_n$ be the maximum number of black squares possible, prove that there exist some real constants $a$, $b$, $c\ge 0$ such that; $$an^2-bn\le d_n\le an^2+cn.$$ [i]Proposed by Ivan Chan Kai Chin[/i]