Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Suppose each element $i\in S=\{1,2,\ldots,n\}$ is assigned a nonempty set $S_i\subseteq S$ so that the following conditions are fulfilled: (i) for any $i,j\in S$, if $j\in S_i$ then $i\in S_j$; (ii) for any $i,j\in S$, if $|S_i|=|S_j|$ then $S_i\cap S_j=\emptyset$. Prove that there exists $k\in S$ for which $|S_k|=1$.

There are $n+1$ containers arranged in a circle. One container has $n$ stones, the others are empty. A move is to choose two containers $A$ and $B$, take a stone from $A$ and put it in one of the containers adjacent to $B$, and to take a stone from $B$ and put it in one of the containers adjacent to $A$. We can take $A = B$. For which $n$ is it possible by series of moves to end up with one stone in each container except that which originally held $n$ stones.

The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers: \[a, b, c, a+b, b+c, c+a, a+b+c\] belong to $S$.

There are $n$ people in line, counting $1,2,\cdots, n$ from left to right, those who count odd numbers quit the line, the remaining people press 1,2 from right to left, and count off again, those who count odd numbers quit the line, and then the remaining people count off again from left to right$\cdots$ Keep doing that until only one person is in the line. $f(n)$ is the number of the last person left at the first count. Find the expression for $f(n)$ and find the value of $f(2022)$

Two players take turns writing on the board in a row from left to right arbitrary numbers. The player loses, after whose move one or more several digits written in a row form a number divisible by $11$. Which player will win if played correctly?

Is it possible to cover a $2003 \times 2003$ chessboard (without overlap) using only horizontal $1 \times 2$ dominoes and only vertical $3 \times 1$ trominoes?

In a school there are totally $n > 2$ classes and not all of them have the same numbers of students. It is given that each class has one head student. The students in each class wear hats of the same color and different classes have different hat colors. One day all the students of the school stand in a circle facing toward the center, in an arbitrary order, to play a game. Every minute, each student put his hat on the person standing next to him on the right. Show that at some moment, there are $2$ head students wearing hats of the same color.

There are $n$ cities on each side of Hung river, with two-way ferry routes between some pairs of cities across the river. A city is “convenient” if and only if the city has ferry routes to all cities on the other side. The river is “clear” if we can find $n$ different routes so that the end points of all these routes include all $2n$ cities. It is known that Hung river is currently unclear, but if we add any new route, then the river becomes clear. Determine all possible values for the number of convenient cities. [i] Proposed by usjl[/i]

How many positive five-digit integers are there that have the product of their five digits equal to $900$? (Karl Czakler)

Let $S$ be a set of $100$ positive integer numbers having the following property: “Among every four numbers of $S$, there is a number which divides each of the other three or there is a number which is equal to the sum of the other three.” Prove that the set $S$ contains a number which divides all other $99$ numbers of $S$. [i]Proposed by Tajikistan[/i]

The hall in a castle is a regular hexagon where its sides' length is 6 meters. The floor of the hall is to be tiled with equilateral triangular tiles where its sides' length is 50 centimeters. Each tile is divided into three congruent triangles by their altitudes up to its orthocenter (see below). Each of these small triangles are colored such that each tile has different colors and no two tiles have identical colorings. How many colors at least are required? A tile's pattern is: [asy] draw((0,0.000)--(2,0.000)); draw((2,0.000)--(1,1.732)); draw((1,1.732)--(0,0.000)); draw((1,0.577)--(0,0.000)); draw((1,0.577)--(2,0.000)); draw((1,0.577)--(1,1.732)); [/asy]

Let $n$ be a positive integer. The numbers $\{1, 2, ..., n^2\}$ are placed in an $n \times n$ grid, each exactly once. The grid is said to be [i]Muirhead-able[/i] if the sum of the entries in each column is the same, but for every $1 \le i,k \le n-1$, the sum of the first $k$ entries in column $i$ is at least the sum of the first $k$ entries in column $i+1$. For which $n$ can one construct a Muirhead-able array such that the entries in each column are decreasing? [i]Proposed by Evan Chen[/i]

A sequence of $A$s and $B$s is called [i]antipalindromic [/i] if writing it backwards, then turning all the $A$s into $B$s and vice versa, produces the original sequence. For example $ABBAAB$ is antipalindromic. For any sequence of $A$s and $B$s we define the cost of the sequence to be the product of the positions of the $A$s. For example, the string $ABBAAB$ has cost $1\cdot 4 \cdot 5 = 20$. Find the sum of the costs of all antipalindromic sequences of length $2020$.

[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same. (a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square. (b) Give an example of a multimagic square. [b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$ Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression. [b]p3.[/b] Let $\lfloor x \rfloor$ be the largest integer that is less than or equal to $x$. For example, $\lfloor 3.9 \rfloor = 3$ and $\lfloor 4\rfloor = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$ [b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible. [b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$ ($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits). [b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table: the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matches. Then they take turns making moves. In a move, a player may take any nonzero number of matches FROM ONE PILE. The player who takes the last match from the table loses the game. a) The player who makes the first move can win the game. What is the winning first move? b) How can he win? (Describe his strategy.) [b]p3.[/b] The planet Naboo is under attack by the imperial forces. Three rebellion camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle with a defensive field. What is the maximal area that they may need to cover? [b]p4.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$ 1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back. [b]p5.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Oleksii and Solomiya play the following game on a square $6n\times 6n$, where $n$ is a positive integer. Oleksii in his turn places a piece of type $F$, consisting of three cells, on the board. Solomia, in turn, after each move of Oleksii, places the numbers $0, 1, 2$ in the cells of the figure that Oleksii has just placed, using each of the numbers exactly once. If two of Oleksii's pieces intersect at any moment (have a common square), he immediately loses. Once the square is completely filled with numbers, the game stops. In this case, if the sum of the numbers in each row and each column is divisible by $3$, Solomiya wins, and otherwise Oleksii wins. Who can win this game if the figure of type $F$ is: a) a rectangle ; b) a corner of three cells? [i]Proposed by Oleksii Masalitin[/i]

Prove the following identity for every $ n\in N$: $ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$

Let $a_n$ be a sequence recursively defined by $a_0 = 0, a_1 = 1$ and $a_{n+2} = a_{n+1} + a_n$. Calculate the sum of $a_n\left( \frac 25\right)^n$ for all positive integers $n$. For what value of the base $b$ we get the sum $1$?

Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$, where the probability is $\tfrac{a}{b}$ and $a$ and $b$ are relatively prime positive integers.

For a pair $ A \equal{} (x_1, y_1)$ and $ B \equal{} (x_2, y_2)$ of points on the coordinate plane, let $ d(A,B) \equal{} |x_1 \minus{} x_2| \plus{} |y_1 \minus{} y_2|$. We call a pair $ (A,B)$ of (unordered) points [i]harmonic[/i] if $ 1 < d(A,B) \leq 2$. Determine the maximum number of harmonic pairs among 100 points in the plane.

Let $n$ be a positive integer. Tasty and Stacy are given a circular necklace with $3n$ sapphire beads and $3n$ turquoise beads, such that no three consecutive beads have the same color. They play a cooperative game where they alternate turns removing three consecutive beads, subject to the following conditions: [list] [*]Tasty must remove three consecutive beads which are turquoise, sapphire, and turquoise, in that order, on each of his turns. [*]Stacy must remove three consecutive beads which are sapphire, turquoise, and sapphire, in that order, on each of her turns. [/list] They win if all the beads are removed in $2n$ turns. Prove that if they can win with Tasty going first, they can also win with Stacy going first. [i]Yannick Yao[/i]

In a circus, there are $n$ clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one particular colour. Find the largest number $n$ of clowns so as to make the ringmaster's order possible.

In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called [b]uniform[/b] if all three of them are friends with each other, or all three of them are enemies. How many uniform triplets are there?

[b]6.1[/b] Three shooters - Anilov, Borisov and Vorobiev - made $6$ each shots at one target and scored equal points. It is known that Anilov scored $43$ points in the first three shots, and Borisov scored $43$ points with the first shot knocked out 3 points. How many points did each shooter score per shot? if there was one hit in 50, two in 25, three in 20, three in 10, two in 5, in 3 - two, in 2 - two, in 1 - three? [img]https://cdn.artofproblemsolving.com/attachments/a/1/4abb71f7bccc0b9d2e22066ec17c31ef139d6a.png[/img] [b]6.2 / 7.4 [/b]Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img]. [b]6.3[/b] The squares of a chessboard contain natural numbers such that each is equal to the arithmetic mean of its neighbors. Sum of numbers standing in the corners of the board is $16$. Find the number standing on the field $e2$. [b]6.4 [/b] There is a table $ 100 \times 100$. What is the smallest number of letters which can be arranged in its cells so that no two are identical the letters weren't next to each other? [b]6.5[/b] The pioneer detachment is lined up in a rectangle. In each rank the tallest is noted, and from these pioneers the most short. In each row, the lowest one is noted, and from them is selected the tallest. Which of these two pioneers is taller? (This means that the two pioneers indicated are the highest of the low and the lowest of tall - must be different) [b]6.6[/b] Find the product of three numbers whose sum is equal to the sum of their squares, equal to the sum of their cubes and equal to $1$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].