Misha has a $100$x$100$ chessboard and a bag with $199$ rooks. In one move he can either put one rook from the bag on the lower left cell of the grid, or remove two rooks which are on the same cell, put one of them on the adjacent square which is above it or right to it, and put the other in the bag. Misha wants to place exactly $100$ rooks on the board, which don't beat each other. Will he be able to achieve such arrangement?

In the table shown in the figure, Zosia replaced eight numbers with their negatives. It turned out that each row and each column contained exactly two negative numbers. Prove that after this change, the sum of all sixteen numbers in the table is equal to $0$. [center] [img] https://wiki-images.artofproblemsolving.com//2/2e/17-3-5.png [/img] [/center]

Let $n$ be a fixed positive odd integer. Take $m+2$ [b]distinct[/b] points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied: 1) $P_0=(0,1),P_{m+1}=(n+1,n)$, and for each integer $i,1\le i\le m$, both $x$- and $y$- coordinates of $P_i$ are integers lying in between $1$ and $n$ ($1$ and $n$ inclusive). 2) For each integer $i,0\le i\le m$, $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and is parallel to the $y$-axis if $i$ is odd. 3) For each pair $i,j$ with $0\le i<j\le m$, line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point. Determine the maximum possible value that $m$ can take.

[b]7.1[/b] Prove that a natural number with an odd number of divisors is a perfect square. [b]7.2[/b] In a triangle $ABC$ with area $S$, medians $AK$ and $BE$ are drawn, intersecting at the point $O$. Find the area of the quadrilateral $CKOE$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png[/img] [b]7.3 .[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires? [b]7.4 [/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]7.5 / 8.4[/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]7.6[/b] Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is $1965$ meters. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

Suppose that $728$ coins are set on a table, all facing heads up at first. For each iteration, we randomly choose $314$ coins and flip them (from heads to tails or vice versa). Let $a/b$ be the expected number of heads after we finish $4001$ iterations, where $a$ and $b$ are relatively prime. Find $a + b$ mod $10000$.

An approximate construction of a regular pentagon goes as follows. Inscribe an arbitrary convex pentagon $P_1P_2P_3P_4P_5$ in a circle. Now choose an arror bound $\epsilon > 0$ and apply the following procedure. (a) Denote $P_0 = P_5$ and $P_6 = P_1$ and construct the midpoint $Q_i$ of the circular arc $P_{i-1}P_{i+1}$ containing $P_i$. (b) Rename the vertices $Q_1,...,Q_5$ as $P_1,...,P_5$. (c) Repeat this procedure until the difference between the lengths of the longest and the shortest among the arcs $P_iP_{i+1}$ is less than $\epsilon$. Prove this procedure must end in a finite time for any choice of $\epsilon$ and the points $P_i$.

For each subset $A$ of $\{1,2, \dots, n \} $, let $M_A$ be the difference between the largest of the elements of $A$ and the smallest of the elements of $A $. Finds the sum of all values ​​of $M_A$ when all possible subsets $A$ of $\{1,2, \dots, n \} $ are considered.

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.

Given is a cube $3 \times 3 \times 3$ with $27$ unit cubes. In each such cube a positive integer is written. Call a $\textit {strip}$ a block $1 \times 1 \times 3$ of $3$ cubes. The numbers are written so that for each cube, its number is the sum of three other numbers, one from each of the three strips it is in. Prove that there are at least $16$ numbers that are at most $60$.

If Percy rolls a fair six-sided die until he rolls a $5$, what is his expected number of rolls, given that all of his rolls are prime?

Given a (simple) graph $G$ with $n \geq 2$ vertices $v_1, v_2, \dots, v_n$ and $m \geq 1$ edges, Joël and Robert play the following game with $m$ coins: [list=i] [*]Joël first assigns to each vertex $v_i$ a non-negative integer $w_i$ such that $w_1+\cdots+w_n=m$. [*]Robert then chooses a (possibly empty) subset of edges, and for each edge chosen he places a coin on exactly one of its two endpoints, and then removes that edge from the graph. When he is done, the amount of coins on each vertex $v_i$ should not be greater than $w_i$. [*]Joël then does the same for all the remaining edges. [*]Joël wins if the number of coins on each vertex $v_i$ is equal to $w_i$. [/list] Determine all graphs $G$ for which Joël has a winning strategy.

Integers are placed on every cell of an infinite checkerboard. For each cell if it contains integer $a$ then the sum of the numbers in the cell under it and the cell right to it is $2a+1$. Prove that in every infinite diagonal row of direction [i] top-right down-left [/i] all numbers are different.

Let $X$ be a set with $n$ elements. Find the largest number of subsets of $X$, each with $3$ elements, so that no two of them are disjoint.

Let $a$ and $b$ be two positive integers and $A$ be a subset of $\{1, 2,…, a + b \}$ that has more than $ \frac {a + b} {2}$ elements. Show that there are two numbers in $A$ whose difference is $a$ or $b$.

There are $12$ points that are vertices of a regular polygon with $12$ sides. Rafael must draw segments that have their two ends at two of the points drawn. He is allowed to have each point be an endpoint of more than one segment and for the segments to intersect, but he is prohibited from drawing three segments that are the three sides of a triangle in which each vertex is one of the $12$ starting points. Find the maximum number of segments Rafael can draw and justify why he cannot draw a greater number of segments.

Ross wants to play solitaire with his deck of $n$ playing cards, but he’s discovered that the deck is “boxed”: some cards are face up, and others are face down. He wants to turn them all face down again, by repeatedly choosing a block of consecutive cards, removing the block from the deck, turning it over, and replacing it back in the deck at the same point. What is the smallest number of such steps Ross needs in order to guarantee that he can turn all the cards face down again, regardless of how they start out?

The central square of the City of Mathematicians is an $n\times n$ rectangular shape, each paved with $1\times 1$ tiles. In order to illuminate the square, night lamps are placed at the corners of the tiles (including the edges of the rectangle) in such a way that each night lamp illuminates all the tiles in its corner. Determine the minimum number of night lamps such that even if one of those night lamps does not work, it is possible to illuminate the entire central square with them.

Let $ n$ be a positive integer. A $4\times n$ rectangular grid is divided in$ 2\times 1$ or $1\times 2$ rectangles (as if it were completely covered with tiles of domino, no overlaps or gaps). Then all the grid points which are vertices of one of the $2\times 1$ or $1\times 2$ rectangles, are painted red. What is the least amount of red points you can get?

Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

2011 numbers are written on a board. For any three numbers, their sum is also among numbers written on the board. What is the smallest number of zeros among all 2011 numbers? (Author: I. Bogdanov)

Let $n>1$ be a positive integer. Claire writes $n$ distinct positive real numbers $x_1, x_2, \dots, x_n$ in a row on a blackboard. In a $\textit{move},$ William can erase a number $x$ and replace it with either $\tfrac{1}{x}$ or $x+1$ at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right. [list] [*]Prove that there exists a positive constant $A,$ independent of $n,$ such that William can always reach his goal in at most $An \log n$ moves. [*]Prove that there exists a positive constant $B,$ independent of $n,$ such that Claire can choose the initial numbers such that William cannot attain his goal in less than $Bn \log n$ moves. [/list]

Let $n$ be a positive integer with $d^2$ positive divisors. We fill a $d\times d$ board with these divisors. At a move, we can choose a row, and shift the divisor from the $i^{\text{th}}$ column to the $(i+1)^{\text{th}}$ column, for all $i=1,2,\ldots, d$ (indices reduced modulo $d$). A configuration of the $d\times d$ board is called [i]feasible[/i] if there exists a column with elements $a_1,a_2,\ldots,a_d,$ in this order, such that $a_1\mid a_2\mid\ldots\mid a_d$ or $a_d\mid a_{d-1}\mid\ldots\mid a_1.$ Determine all values of $n$ for which ragardless of how we initially fill the board, we can reach a feasible configuration after a finite number of moves.

A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies \[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \] for all $k=1,2,\ldots 9$. Find the number of interesting numbers.

In a garden, which is organized as a $2024\times 2024$ board, we plant three types of flowers: roses, daisies, and orchids. We want to plant flowers such that the following conditions are satisfied: (i) Each grid is planted with at most one type of flower. Some grids can be left blank and not planted. (ii) For each planted grid $A$, there exist exactly $3$ other planted grids in the same column or row such that those $3$ grids are planted with flowers of different types from $A$'s. (iii) Each flower is planted in at least $1$ grid. What is the maximal number of the grids that can be planted with flowers?