In a contest, there are $11$ contestants to solve $9$ math problems. After the end of the contest, it was found that any two contestants solved no more than $ 1$ problem together. Find the largest positive integer $k$ such that each problem can be solved by at least $k$ candidates.

A [i]lame king[/i] is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7\times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that [b](a)[/b] The sum of the elements of $A$ is $0,$ [b](b)[/b] For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$. Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that \[|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.\] Where $|X|$ denote the numbers of the elements in set $X$.

Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $ a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$, Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$. [i]Proposed by Navid Safaei [/i]

The set of integers is coloured in $100$ colours in such a way that all the colours are used and the following is true. For any choice of intervals $[a, b]$ and $[c,d]$ of equal length and with integral endpoints, if a and c as well as $b$ and $d$, respectively, have the same colour, then the whole intervals $[a, b]$ and $[c,d]$ are identically coloured in that, for any integer $x$, $0 \le x \le b - a$, the numbers $a + x$ and $c + x$ are of the same colour. Prove that $-1982$ and $1982$ are of different colours

The numbers $1, 2, 3, ..., 99, 100$ are randomly arranged in the cells of a square table measuring $10\times 10$ (each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 1$82$. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.

A country has $2023$ cities and there are flights between these cities. Each flight connects two cities in both directions. We know that you can get from any city to any other using these flights, and from each city there are flights to at most $4$ other cities. What is the maximum possible number of cities in the country from which there is a flight to only one city?

A convex $2019$-gon $A_1A_2...A_{2019}$ is cut into smaller pieces along its $2019$ diagonals of the form $A_iA_{i+3}$ for $1 \le i \le2019$, where $A_{2020} = A_1$, $A_{2021} = A_2$, and $A_{2022} = A_3$. What is the least possible number of resulting pieces?

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

Initially, a word of $250$ letters with $125$ letters $A$ and $125$ letters $B$ is written on a blackboard. In each operation, we may choose a contiguous string of any length with equal number of letters $A$ and equal number of letters $B$, reverse those letters and then swap each $B$ with $A$ and each $A$ with $B$ (Example: $ABABBA$ after the operation becomes $BAABAB$). Decide if it possible to choose initial word, so that after some operations, it will become the same as the first word, but in reverse order.

Lucky starts doodling on a $5\times 5$ Bingo board. He puts his pencil at the center of the upper-left square (marked by ‘·’) and draws a continuous doodle ending on the Free Space, never going off the board or through a corner of a square. (See Figure 1.) (a) Is it possible for Lucky’s doodle to visit all squares exactly once? (The starting and ending squares are considered visited.) (b) Is it possible for Lucky’s doodle to visit all squares exactly twice?

A cube of side length $n$ is made up of $n^3$ smaller unit cubes. Some of the six faces of the large cube are fully painted. When the large cube is taken apart, 245 smaller cubes do not have any paint on them. Determine the value(s) of $n$ and how many faces of the large cube were painted.

Numbers $m$ and $n$ are given positive integers. There are $mn$ people in a party, standing in the shape of an $m\times n$ grid. Some of these people are police officers and the rest are the guests. Some of the guests may be criminals. The goal is to determine whether there is a criminal between the guests or not.\\ Two people are considered \textit{adjacent} if they have a common side. Any police officer can see their adjacent people and for every one of them, know that they're criminal or not. On the other hand, any criminal will threaten exactly one of their adjacent people (which is likely an officer!) to murder. A threatened officer will be too scared, that they deny the existence of any criminal between their adjacent people.\\ Find the least possible number of officers such that they can take position in the party, in a way that the goal is achievable. (Note that the number of criminals is unknown and it is possible to have zero criminals.) [i]Proposed by Abolfazl Asadi[/i]

Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.

assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.

[b]6.1[/b] The student bought a briefcase, a fountain pen and a book. If the briefcase cost 5 times cheaper, the fountain pen was 2 times cheaper, and the book was 2 1/2 times cheaper cheaper, then the entire purchase would cost 2 rubles. If the briefcase was worth 2 times cheaper, a fountain pen is 4 times cheaper, and a book is 3 times cheaper, then the whole the purchase would cost 3 rubles. How much does it really cost? ´ [b]6.2.[/b] Which number is greater: $$\underbrace{888...88}_{19 \, digits} \cdot \underbrace{333...33}_{68 \, digits} \,\,\, or \,\,\, \underbrace{444...44}_{19 \, digits} \cdot \underbrace{666...67}_{68 \, digits} \, ?$$ [b]6.3[/b] Distance between Luga and Volkhov 194 km, between Volkhov and Lodeynoye Pole 116 km, between Lodeynoye Pole and Pskov 451 km, between Pskov and Luga 141 km. What is the distance between Pskov and Volkhov? [b]6.4 [/b] There are $4$ objects in pairs of different weights. How to use a pan scale without weights Using five weighings, arrange all these objects in order of increasing weights? [b]6.5 [/b]. Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. [b]6.6 [/b] In task 6.1, determine what is more expensive: a briefcase or a fountain pen. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

Let $X$ be a finite set of real numbers. For any $x,x' \in X$ with $x<x'$, define a function $f(x,x')$, then $f$ is called an ordered pair function on $X$. For any given ordered pair function $f$ on $X$, if there exist elements $x_1 <x_2 <\cdots<x_k$ in $X$ such that $f(x_1 ,x_2 ) \le f(x_2 ,x_3 ) \le \cdots \le f(x_{k-1} ,x_k )$, then $x_1 ,x_2 ,\cdots,x_k$ is called an $f$-ascending sequence of length $k$ in $X$. Similarly, define an $f$-descending sequence of length $l$ in $X$. For integers $k,l \ge 3$, let $h(k,l)$ denote the smallest positive integer such that for any set $X$ of $s$ real numbers and any ordered pair function $f$ on $X$, there either exists an $f$-ascending sequence of length $k$ in $X$ or an $f$-descending sequence of length $l$ in $X$ if $s \ge h(k,l)$. Prove: 1.For $k,l>3,h(k,l) \le h(k-1,l)+h(k,l-1)-1$; 2.$h(k,l) \le \binom{l-2}{k+l-4} +1$.

Given an integer $ n > 3.$ Prove that there exists a set $ S$ consisting of $ n$ pairwisely distinct positive integers such that for any two different non-empty subset of $ S$:$ A,B, \frac {\sum_{x\in A}x}{|A|}$ and $ \frac {\sum_{x\in B}x}{|B|}$ are two composites which share no common divisors.

In the kindergarten there is a big box with balls of three colors: red, blue and green, 100 balls in total. Once Pasha took out of the box 30 red, 10 blue, and 20 green balls and played with them. Then he lost five balls and returned the others back into the box. The next day, Sasha took out of the box 8 red, 18 blue, and 48 green balls. Is it possible to determine the color of at least one lost ball?

Let $n$ be a positive integer. On a board there are written all integers from $1$ to $n$. Alina does $n$ moves consecutively: for every integer $m$ $(1 \leq m \leq n)$ the move $m$ consists in changing the sign of every number divisible by $m$. At the end Alina sums the numbers. Find this sum.

Determine the maximum number of bishops that we can place in a $8 \times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop. Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the $2$ main diagonals and the ones parallel to those ones.

[b]p1.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form a strictly increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.) [b]p2.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that $$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B1B1B1...}) + 1$$ Find the sum of all possible values of $10A + B$. [b]p3.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $\overline{AC}$ and $\overline{BC}$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE =EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a$, $b$ positive integers and $a$ squarefree, then find $a + b$. [b]p4.[/b] Five bowling pins $P_1, P_2, ..., P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is $\frac{a}{b}$ where $a$ and $b$ are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i - j| = 1$.) [b]p5.[/b] How many terms in the expansion of $$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$ have coeffcients equal to $1011$? [b]p6.[/b] Suppose $f(x)$ is a monic quadratic polynomial with distinct nonzero roots $p$ and $q$, and suppose $g(x)$ is a monic quadratic polynomial with roots $p + \frac{1}{q}$ and $q + \frac{1}{p}$ . If we are given that $g(-1) = 1$ and $f(0)\ne -1$, then there exists some real number $r$ that must be a root of $f(x)$. Find $r$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a $4\times 4$ array of pairwise distinct positive integers such that on each column, respectively row, one of the numbers is equal to the sum of the other three. Determine the least possible value of the largest number such an array may contain. [i]The Problem Selection Committee[/i]

A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers $ n$ for which it is possible to build a cube of side $ n$ using such bricks.