Each number from the set $\{1,2,3,4,5,6,7,8\}$ is either colored red or blue, following these rules: a) The number $4$ is colored red, and there is at least one number colored blue. b) If two numbers $x, y$ have different colors and $x + y \leq 8$, then the number $x + y$ is colored blue. c) If two numbers $x, y$ have different colors and $x \cdot y \leq 8$, then the number $x \cdot y$ is colored red. Find all possible ways the numbers from this set can be colored.

Let $P$ be a convex polygon, and let $n \ge 3$ be a positive integer. On each side of $P$, erect a regular $n$-gon that shares that side of $P$, and is outside $P$. If none of the interiors of these regular n-gons overlap, we call P $n$-[i]good[/i]. (a) Find the largest value of $n$ such that every convex polygon is $n$-[i]good[/i]. (b) Find the smallest value of $n$ such that no convex polygon is $n$-[i]good[/i].

In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.

We play a game of musical chairs with $n$ chairs numbered $1$ to $n$. You attach $n$ leaves, numbered $1$ to $n$, to the chairs in such a way that the number on a leaf does not match the number on the chair it is attached to. One player sits on each chair. Every time you clap, each player looks at the number on the leaf attached to his current seat and moves to sit on the seat with that number. Prove that, for any $m$ that is not a prime power with$ 1 < m \leq n$, it is possible to attach the leaves to the seats in such a way that after $m$ claps everyone has returned to the chair they started on for the first time.

Let $N$ be a positive integer. Initially, a positive integer $A$ is written on the board. At each step, we can perform one of the following two operations with the number written on the board: (i) Add $N$ to the number written on the board and replace that number with the sum obtained; (ii) If the number on the board is greater than $1$ and has at least one digit $1$, then we can remove the digit $1$ from that number, and replace the number initially written with this one (with removal of possible leading zeros) For example, if $N = 63$ and $A = 25$, we can do the following sequence of operations: $$25 \rightarrow 88 \rightarrow 151 \rightarrow 51 \rightarrow 5$$ And if $N = 143$ and $A = 2$, we can do the following sequence of operations: $$2 \rightarrow 145 \rightarrow 288 \rightarrow 431 \rightarrow 574 \rightarrow 717 \rightarrow 860 \rightarrow 1003 \rightarrow 3$$ For what values of $N$ is it always possible, regardless of the initial value of $A$ on the blackboard, to obtain the number $1$ on the blackboard, through a finite number of operations?

Ten closed intervals, each of length $1$, are placed in the interval $[0,4]$. Show that there is a point in the larger interval that belongs to at least four of the smaller intervals.

The cells of a $8 \times 8$ table are colored either black or white so that each row has a different number of black squares, and each column has a different number of black squares. What is the maximum number of pairs of adjacent cells of different colors?

[b]p1.[/b] How many ways can we arrange the elements $\{1, 2, ..., n\}$ to a sequence $a_1, a_2, ..., a_n$ such that there is only exactly one $a_i$, $a_{i+1}$ such that $a_i > a_{i+1}$? [b]p2. [/b]How many distinct (non-congruent) triangles are there with integer side-lengths and perimeter $2012$? [b]p3.[/b] Let $\phi$ be the Euler totient function, and let $S = \{x| \frac{x}{\phi (x)} = 3\}$. What is $\sum_{x\in S} \frac{1}{x}$? [b]p4.[/b] Denote $f(N)$ as the largest odd divisor of $N$. Compute $f(1) + f(2) + f(3) +... + f(29) + f(30)$. [b]p5.[/b] Triangle $ABC$ has base $AC$ equal to $218$ and altitude $100$. Squares $s_1, s_2, s_3, ...$ are drawn such that $s_1$ has a side on $AC$ and has one point each touching $AB$ and $BC$, and square $s_k$ has a side on square $s_{k-}1$ and also touches $AB$ and $BC$ exactly once each. What is the sum of the area of these squares? [b]p6.[/b] Let $P$ be a parabola $6x^2 - 28x + 10$, and $F$ be the focus. A line $\ell$ passes through $F$ and intersects the parabola twice at points $P_1 = (2,-22)$, $P_2$. Tangents to the parabola with points at $P_1, P_2$ are then drawn, and intersect at a point $Q$. What is $m\angle P_1QP_2$? PS. You had better use hide for answers.

A word of length $n$ is an ordered sequence $x_1x_2\ldots x_n$ where $x_i$ is a letter from the set $\{ a,b,c \}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2,$ contains all the letters $a,b,c$. Prove that $|B_{n+1}|=3|A_n|$. [i]Vasile Pop[/i]

You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots, a_k$. It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there are exactly $a_1$ numbers of the first color, $a_2$ numbers of the second color, $\ldots$, and $a_k$ numbers of the $k$-th color, there is always a number $x \in \{1, 2, \ldots, 2021\}$, such that the total number of numbers colored in the same color as $x$ is exactly $x$. What are the possible values of $k$? [i]Proposed by Arsenii Nikolaiev[/i]

In a school there are $2007$ male and $2007$ female students. Each student joins not more than $100$ clubs in the school. It is known that any two students of opposite genders have joined at least one common club. Show that there is a club with at least $11$ male and $11$ female members.

Let $n$ be a positive integer. Each point $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers with $x+y<n$, is coloured red or blue, subject to the following condition: if a point $(x,y)$ is red, then so are all points $(x',y')$ with $x'\leq x$ and $y'\leq y$. Let $A$ be the number of ways to choose $n$ blue points with distinct $x$-coordinates, and let $B$ be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$.

On the ends of the diameter two "$1$"s are written. Each of the semicircles is divided onto two parts and the sum of the numbers of its ends (i.e. "$2$") is written at the midpoint. Then every of the four arcs is halved and in its midpoint the sum of the numbers on its ends is written. Find the total sum of the numbers on the circumference after $n$ steps.

Prove that there exists a partition of the set $A = \{1^3, 2^3, \ldots , 2000^3\}$ into $19$ nonempty subsets such that the sum of elements of each subset is divisible by $2001^2$.

There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week.

Divide the upper right quadrant of the plane into square cells with side length $1$. In this quadrant, $n^2$ cells are colored, show that there’re at least $n^2+n$ cells (possibly including the colored ones) that at least one of its neighbors are colored.

On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

Kolya and Dima play a game on an $8\times 8$ board, making moves in turn. During his turn, Kolya must put one cross in any empty cell (i.e., in a cell in which a cross has not yet been drawn and which has not yet been covered with a domino). Dima must cover two adjacent cells with a domino (which are not yet covered with other dominoes), in which there are an even number of crosses in total (0 or 2). The one who can't make a move loses. Which of does the player have a winning strategy, if [list=a] [*]Dima makes the first move? [*]Kolya makes the first move? [/list] [i]Proposed by M. Didin[/i]

On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial on the blackboard has $37$ distinct positive roots. [i](8 points)[/i] [i]Alexandr Kuznetsov[/i]

Let $ A_0 \equal{} (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k \equal{} (x_1,\dots,x_k)$ we construct a new sequence $ A_{k \plus{} 1}$ in the following way. 1. We choose a partition $ \{1,\dots,n\} \equal{} I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression \[ \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right| \] attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily. 2. We set $ A_{k \plus{} 1} \equal{} (y_1,\dots,y_n)$ where $ y_i \equal{} x_i \plus{} 1$ if $ i\in I$, and $ y_i \equal{} x_i \minus{} 1$ if $ i\in J$. Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$. [i]Author: Omid Hatami, Iran[/i]

Consider the regular $101$-gon. A line $l$ does not contain any vertex of this polygon. Prove that line $l$ intersects even number of the diagonals of this polygon.

$N$ oligarchs built a country with $N$ cities with each one of them owning one city. In addition, each oligarch built some roads such that the maximal amount of roads an oligarch can build between two cities is $1$ (note that there can be more than $1$ road going through two cities, but they would belong to different oligarchs). A total of $d$ roads were built. Some oligarchs wanted to create a corporation by combining their cities and roads so that from any city of the corporation you can go to any city of the corporation using only corporation roads (roads can go to other cities outside corporation) but it turned out that no group of less than $N$ oligarchs can create a corporation. What is the maximal amount that $d$ can have?

Show that there exists a proper non-empty subset $S$ of the set of real numbers such that, for every real number $x$, the set $\{nx + S : n \in N\}$ is finite, where $nx + S =\{nx + s : s \in S\}$

The cells of an $n \times n$ table are filled with the numbers $1,2,\dots,n$ for the first row, $n+1,n+2,\dots,2n$ for the second, and so on until $n^2-n,n^2-n+1,\dots,n^2$ for the $n$-th row. Peter picks $n$ numbers from this table such that no two of them lie on the same row or column. Peter then calculates the sum $S$ of the numbers he has chosen. Prove that Peter always gets the same number for $S$, no matter how he chooses his $n$ numbers.