Let $Q$ be a $(2n+1) \times (2n+1)$ board. Some of its cells are colored black in such a way that every $2 \times 2$ board of $Q$ has at most $2$ black cells. Find the maximum amount of black cells that the board may have.

A toy factory produces several kinds of clay toys. The toys are painted in $k$ colours. [i]Diversity[/i] of a colour is the number of [i]different[/i] toys of that colour. (Thus, if there are $5$ blue cats, $7$ blue mice and nothing else is blue, the diversity of colour blue is $2$.) The painting protocol requires that [i]each colour is used and the diversities of each two colours are different[/i]. The toys in the store could be painted according to the protocol. However, a batch of clay Cheburashkas arrived at the store before painting (there were no Cheburashkas before). The number of Cheburashkas is not less that the number of the toys of any other kind. The total number of all toys, including Cheburashkas, is at least $\frac{(k+1)(k+2)}{2}$. Prove that now the toys can be painted in $k + 1$ colours according to the protocol. [i]Proposed by F. Petrov[/i]

Let $n,s,$ and $t$ be positive integers and $0<\lambda<1.$ A simple graph on $n$ vertices with at least $\lambda n^2$ edges is given. We say that $(x_1,\ldots,x_s,y_1,\ldots,y_t)$ is a [i]good intersection[/i] if letters $x_i$ and $y_j$ denote not necessarily distinct vertices and every $x_iy_j$ is an edge of the graph $(1\leq i\leq s,$ $1\leq j\leq t).$ Prove that the number of good insertions is at least $\lambda^{st}n^{s+t}.$ [i]Proposed by Kada Williams, Cambridge[/i]

p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday $4$th March $2013$ at $10.00$ am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower. March $6$, $2013$ the clock is late six minutes compared to Jam Gadang. The following days Bahri observed that his antique clock exhibited the same pattern of delay. On what day and what date in $2014$ the antique Bahri clock (hand short and long hands) point to the same number as the Clock Tower? p2. In one season, the Indonesian Football League is participated by $20$ teams football. Each team competes with every other team twice. The result of each match is $3$ if you win, $ 1$ if you draw, and $0$ if you lose. Every week there are $10$ matches involving all teams. The winner of the competition is the team that gets the highest total score. At the end what week is the fastest possible, the winner of the competition on is the season certain? p3. Look at the following picture. The quadrilateral $ABCD$ is a cyclic. Given that $CF$ is perpendicular to $AF$, $CE$ is perpendicular to $BD$, and $CG$ is perpendicular to $AB$. Is the following statements true? Write down your reasons. $$\frac{BD}{CE}=\frac{AB}{CG}+ \frac{AD}{CF}$$ [img]https://cdn.artofproblemsolving.com/attachments/b/0/dbd97b4c72bc4ebd45ed6fa213610d62f29459.png[/img] p4. Suppose $M=2014^{2014}$. If the sum of all the numbers (digits) that make up the number $M$ equals $A$ and the sum of all the digits that make up the number $A$ equals $B$, then find the sum of all the numbers that make up $B$. p5. Find all positive integers $n < 200$ so that $n^2 + (n + 1)^2$ is square of an integer.

Let $ 0 < a_1 < a_2 < a_3 < ... < a_{10000} < 20000$ be integers such that $ gcd(a_i,a_j) < a_i, \forall i < j$ ; is $ 500 < a_1$ [i](always)[/i] true ? [i](own)[/i] :lol:

The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that a) there are three consecutive numbers with the sum being at least $15$, b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.

Let $B$ be a set of more than $\tfrac{2^{n+1}}{n}$ distinct points with coordinates of the form $(\pm 1, \pm 1, \cdots, \pm 1)$ in $n$-dimensional space with $n \ge 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle.

The signs "$+$" or "$-$" are placed in all cells of a $4 \times 4$ square table. It is allowed to change a sign of any cell altogether with signs of all its adjacent cells (i.e. cells having a common side with it). Find the number of different tables that could be obtained by iterating this procedure.

Csongi bought a $12$-sided convex polygon-shaped pizza. The pizza has no interior point with three or more distinct diagonals passing through it. Áron wants to cut the pizza along $3$ diagonals so that exactly $6$ pieces of pizza are created. In how many ways can he do this? Two ways of slicing are different if one of them has a cut line that the other does not have.

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

Let $\mathbb Z_{\ge 0}$ be the set of non-negative integers, and let $f:\mathbb Z_{\ge 0}\times \mathbb Z_{\ge 0} \to \mathbb Z_{\ge 0}$ be a bijection such that whenever $f(x_1,y_1) > f(x_2, y_2)$, we have $f(x_1+1, y_1) > f(x_2 + 1, y_2)$ and $f(x_1, y_1+1) > f(x_2, y_2+1)$. Let $N$ be the number of pairs of integers $(x,y)$ with $0\le x,y<100$, such that $f(x,y)$ is odd. Find the smallest and largest possible values of $N$.

There are $90$ cards and two different digits are written on each one: $01$, $02$, $03$, $04$, $05$, $06$, $07$, $08$, $09$, $10$, $12$, and so on up to $98$. A set of cards is [i]correct [/i]if it does not contain any cards whose first digit is the same as the second digit of another card in the set. We call the [i]value [/i]of a set of cards the sum of the numbers written on each card. For example, the four cards $04$, $35$, $78$ and $98$ form a correct set and their value is $215$, since$ 04+35+78+98=215$. Find a correct set that has the largest possible value. Explain why it is impossible to achieve a correct set of higher value.

A tournament is held among $n$ teams, following such rules: a) every team plays all others once at home and once away.(i.e. double round-robin schedule) b) each team may participate in several away games in a week(from Sunday to Saturday). c) there is no away game arrangement for a team, if it has a home game in the same week. If the tournament finishes in 4 weeks, determine the maximum value of $n$.

Let $n$ be a natural number. We define $f(n)$ as the number of ways to express $n$ as a sum of powers of $2$, where the order of the terms is taken into account. For example, $f(4) = 6$, because $4$ can be written as: \begin{align*} 4;\\ 2 + 2;\\ 2 + 1 + 1;\\ 1 + 2 + 1;\\ 1 + 1 + 2;\\ 1 + 1 + 1 + 1. \end{align*} Find the smallest $n$ greater than $2019$ for which $f(n)$ is odd.

Let $n$ and $k$ be integers such that $0\leq k\leq n$. Prove that $\sum_{u=0}^{k}\binom{n+u-1}{u}\binom{n}{k-2u}=\binom{n+k-1}{k}$. Note that we use the following conventions: $\binom{r}{0}=1$ for every integer $r$; $\binom{u}{v}=0$ if $u$ is a nonnegative integer and $v$ is an integer satisfying $v<0$ or $v>u$. Darij

For which positive integers $n\geq4$ does there exist a convex $n$-gon with side lengths $1, 2, \dots, n$ (in some order) and with all of its sides tangent to the same circle?

Three non-negative integers are written on the board. In every step, the three numbers $(a, b, c)$ are being replaced with $a+b, b+c, c+a$. Find the biggest number of steps, after which the number $111$ will appear on the board.

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

Let p be a prime and $a_1 , a_2 , ..., a_k$ pairwise incongruent modulo p . Prove that $[\sqrt {k-1}]$ of the elements can be selected from $a_i$'s such that adding any numbers different from the selected ones will never give a number divisible by p .

There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?

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 $10000$.

Let $m,n\geq 2$ be positive integers and $S\subseteq [1,m]\times [1,n]$ be a set of lattice points. Prove that if \[|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloor\]then there exists a circle which passes through at least four distinct points of $S.$

Let $T$ be a $2015 \times 2015$ array containing the integers $1, 2, 3, ... , 2015^2$ satisfying the property that $T_{i,a }> T_{i,b}$ for all $a > b$ and $T_{c,j} > T_{d,j}$ for all $c > d$ where $1 \le a, b, c, d \le 2015$ and $T_{i,j}$ represents the entry in the $i$-th row and $j$-th column of $T$. How many possible values are there for the entry at $T_{5,5}$?

In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.

$16$ children are sitting at a round table. After the break, they sit down again on table. They find that each child is either sitting on its original [lace or in one of the two neighboring places. How many seating arrangements are possible in this way after the break?