Find the smallest positive integer $k$ with the following property: if each cell of a $100\times 100$ grid is dyed with one color and the number of cells of each color is not more than $104$, then there is a $k\times1$ or $1\times k$ rectangle that contains cells of at least three different colors.

A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.

[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$. [b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$. [b]p3.[/b] For a positive integer $n$, there is a sequence $(a_0, a_1, a_2,..., a_n)$ of real values such that $a_0 = 11$ and $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$ for every $k$ with $0 \le k \le n-1$. Find the maximum possible value of $n$. (Be careful that your answer isn’t off by one!) [b]p4.[/b] Persons $A$ and $B$ stand at point $P$ on line $\ell$. Point $Q$ lies at a distance of $10$ from point $P$ in the direction perpendicular to $\ell$. Both persons intially face towards $Q$. Person $A$ walks forward and to the left at an angle of $25^o$ with $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the right, and continues walking. Person $B$ walks forward and to the right at an angle of $55^o$ with line $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the left, and continues walking. Their paths cross at point $R$. Find the distance $PR$. [b]p5.[/b] Compute $$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$ [b]p6.[/b] There is a unique real value $A$ such that for all $x$ with $1 < x < 3$ and $x \ne 2$, $$\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.$$ Compute $A$. [b]p7.[/b] Nine poles of height $1, 2,..., 9$ are placed in a line in random order. A pole is called [i]dominant [/i] if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly $2$ dominant poles. [b]p8.[/b] $\tan (11x) = \tan (34^o)$ and $\tan (19x) = \tan (21^o)$. Compute $\tan (5x)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the maximum positive integer $k$ such that there exist $k$ positive integers which do not exceed $2017$ and have the property that every number among them cannot be a power of any of the remaining $k-1$ numbers.

Let $n\in \mathbb{Z}_{> 0}$. The set $S$ contains all positive integers written in decimal form that simultaneously satisfy the following conditions: [list=1][*] each element of $S$ has exactly $n$ digits; [*] each element of $S$ is divisible by $3$; [*] each element of $S$ has all its digits from the set $\{3,5,7,9\}$ [/list] Find $\mid S\mid$

Let $n$ and $k$ be positive integers, with $n\geq k+1$. There are $n$ countries on a planet, with some pairs of countries establishing diplomatic relations between them, such that each country has diplomatic relations with at least $k$ other countries. An evil villain wants to divide the countries, so he executes the following plan: (1) First, he selects two countries $A$ and $B$, and let them lead two allies, $\mathcal{A}$ and $\mathcal{B}$, respectively (so that $A\in \mathcal{A}$ and $B\in\mathcal{B}$). (2) Each other country individually decides wether it wants to join ally $\mathcal{A}$ or $\mathcal{B}$. (3) After all countries made their decisions, for any two countries with $X\in\mathcal{A}$ and $Y\in\mathcal{B}$, eliminate any diplomatic relations between them. Prove that, regardless of the initial diplomatic relations among the countries, the villain can always select two countries $A$ and $B$ so that, no matter how the countries choose their allies, there are at least $k$ diplomatic relations eliminated. [i]Proposed by YaWNeeT.[/i]

The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$. After the competition, they queue in front of the restaurant according to the following rules. [list] [*]The Jury chooses the initial order of the contestants in the queue. [*]Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$. [list] [*]If contestant $C_i$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions. [*]If contestant $C_i$ has fewer than $i$ other contestants in front of her, the restaurant opens and process ends. [/list] [/list] [list=a] [*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices. [*]Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves. [/list]

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$. [i]Brian Hamrick.[/i]

In the plane, there is a figure in the form of an $L$-tromino, which is composed of $3$ unit squares, which we will denote by $\Phi_0$. On every move, we choose an arbitrary straight line in the plane and using it we construct a new figure. The $\Phi_n$, obtained in the $n$-th move, is obtained as the union of the figure $\Phi_{n-1}$ and its axial reflection with respect to the chosen line. Also, for the move to be valid, it is necessary that the surface of the newly obtained piece to be twice as large as the previous one. Is it possible to cover the whole plane in that process?

Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$. [i] (From "Radu Păun" contest, Radu Miculescu)[/i]

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many parts can space be divided into by : a) three half-plane? b) four half-planes?

The wizards $A, B, C, D$ know that the integers $1, 2, \ldots, 12$ are written on 12 cards, one integer on each card, and that each wizard will get three cards and will see only his own cards. Having received the cards, the wizards made several statements in the following order. [list=A] [*]“One of my cards contains the number 8”. [*]“All my numbers are prime”. [*]“All my numbers are composite and they all have a common prime divisor”. [*]“Now I know all the cards of each wizard”. [/list] What were the cards of $A{}$ if everyone was right? [i]Mikhail Evdokimov[/i]

Is it possible to arrange $n \times n$ in the cells of a square table the numbers $0$,$ 1$ or $2$ so that the sums of the numbers in rows and columns took on all different values from $1$ to $2n$? Consider two cases: a) $n$ is an odd number; b) $n$ is an even number.

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]

In a group of interpreters each one speaks one of several foreign languages, 24 of them speak Japanese, 24 Malaysian, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malaysian and exactly 12 speak Farsi.

Let $a_1, a_2, \ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of $1,2, \ldots ,2007$. Define an [b]operation[/b] to be 22 consecutive applications of the following steps on the sequence $S$: on $i$-th step, choose a number from the sequense $S$ at random, say $x$. If $1 \leq i \leq 11$, replace $x$ with $x+a_i$ ; if $12 \leq i \leq 22$, replace $x$ with $x-a_{i-11}$ . If the result of [b]operation[/b] on the sequence $S$ is an odd permutation of $\{1, 2, \ldots , 2007\}$, it is an [b]odd operation[/b]; if the result of [b]operation[/b] on the sequence $S$ is an even permutation of $\{1, 2, \ldots , 2007\}$, it is an [b]even operation[/b]. Which is larger, the number of odd operation or the number of even permutation? And by how many? Here $\{x_1, x_2, \ldots , x_{2007}\}$ is an even permutation of $\{1, 2, \ldots ,2007\}$ if the product $\prod_{i > j} (x_i - x_j)$ is positive, and an odd one otherwise.

A board $1$x$k$ is called [i]guayaco[/i] if: -Each unit square is painted with exactly one of $k$ available colors. -If $gcd(i,k)>1$, the $i$th unit square is painted with the same color as $(i-1)$th unit square. -If $gcd(i, k)=1$, the $i$th unit square is painted with the same color as $(k-i)$th unit square. Sebastian chooses a positive integer $a$ and calculates the number of boards $1$x$a$ that are guayacos. After that, David chooses a positive integer $b$ and calculates the number of boards $1$x$b$ that are guayacos. David wins if the number of boards $1$x$a$ that are guayacos is the same as the number of boards $1$x$b$ that are guayacos, otherwise, Sebastian wins. Find all the pairs $(a,b) $ such that, with those numbers, David wins.

Find all positive integers $n$ such that one can place checkers on a $n\times n$ checkerboard such that any square chosen from the checkerboard has exactly $2$ adjacent squares with checkers on it. Two squares are considered adjacent if they both share a common side

Let $ G$ be a simple graph with vertex set $ V\equal{}\{0,1,2,3,\cdots ,n\plus{}1\}$ .$ j$and$ j\plus{}1$ are connected by an edge for $ 0\le j\le n$. Let $ A$ be a subset of $ V$ and $ G(A)$ be the induced subgraph associated with $ A$. Let $ O(G(A))$ be number of components of $ G(A)$ having an odd number of vertices. Let $ T(p,r)\equal{}\{A\subset V \mid 0.n\plus{}1 \notin A,|A|\equal{}p,O(G(A))\equal{}2r\}$ for $ r\le p \le 2r$. Prove That $ |T(p,r)|\equal{}{n\minus{}r \choose{p\minus{}r}}{n\minus{}p\plus{}1 \choose{2r\minus{}p}}$.

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

Let $k$ be a positive integer. Some of the $2k$-element subsets of a given set are marked. Suppose that for any subset of cardinality less than or equal to $(k+1)^2$ all the marked subsets contained in it (if any) have a common element. Show that all the marked subsets have a common element.

A museum wants to protect its most valuable piece by maintaining constant surveillance. To do this, he wants to place guards to watch the place, in shifts of $7$ consecutive hours. Each guard starts his shift at the same time every day. A guard is essential if there is any time during the day when you are alone to watch the item. Indicates all possibilities for the number of guards guarding the piece, so that everyone is indispensable.

Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?