Given 98 points in a circle. Mary and Joseph play alternatively in the next way: - Each one draw a segment joining two points that have not been joined before. The game ends when the 98 points have been used as end points of a segments at least once. The winner is the person that draw the last segment. If Joseph starts the game, who can assure that is going to win the game.

Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.

For what values of $k\ge2$ can the set of natural numbers be colored in $k$ colors in such a way that it contains no single - color infinite arithmetic progression, but for any two colors there is a progression whose members are each colored in one of these two colors?

The Sultan gathered $300$ court sages and offered them a test. There are caps of $25$ different colors, known in advance to the sages. The Sultan said that one of these caps will be put on each of the sages, and if for each color write the number of caps worn, then all numbers will be different. Every sage can see the caps of the other sages, but not own cap. Then all the sages will simultaneously announce the supposed color of their cap. Can sages advance agree to act in such a way that at least $150$ of them are guaranteed to name a color right?

A student divides all $30$ marbles into $5$ boxes numbered $1, 2, 3, 4, 5$ (after being divided, there may be a box with no marbles). a) How many ways are there to divide marbles into boxes (are two different ways if there is a box with a different number of marbles)? b) After dividing, the student paints those $30$ marbles by a number of colors (each with the same color, one color can be painted for many marbles), so that there are no $2$ marbles in the same box. have the same color and from any $2$ boxes it is impossible to choose $8$ marbles painted in $4$ colors. Prove that for every division, the student must use no less than $10$ colors to paint the marbles. c) Show a division so that with exactly $10$ colors the student can paint the marbles that satisfy the conditions in question b).

Given an infinite sequence of numbers $a_1, a_2,...$, in which there are no two equal members. Segment $a_i, a_{i+1}, ..., a_{i+m-1}$ of this sequence is called a monotone segment of length $m$, if $a_i < a_{i+1} <...<a_{i+m-1}$ or $a_i > a_{i+1} >... > a_{i+m-1}$. It turned out that for each natural $k$ the term $a_k$ is contained in some monotonic segment of length $k + 1$. Prove that there exists a natural $N$ such that the sequence $a_N , a_{N+1} ,...$ monotonic.

Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime.

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]

Square table $5 \times 5$ is filled with numbers in a following way. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8zLzQ0Y2M1NjdiNjQ3NjhlYTAwMWQ0MTg2ZjIwZWE4NzkwYzcwYWFkLnBuZw==&rn=dGFiZWxpY2EucG5n[/img] We can change the table in a way we take two arbitrary numbers from the table and we decrease both of them with value of smaller of those two. Can we get to the table with all zeros?

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. Compute the real value of $a$ such that $$\sqrt{a(101b + 1)} - 1 = \sqrt{b(c - 1)}+ 10\sqrt{(a - c)b}.$$ [b]p20.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. For some triangle $\vartriangle ABC$, let $\omega$ and $\omega_A$ be the incircle and $A$-excircle with centers $I$ and $I_A$, respectively. Suppose $AC$ is tangent to $\omega$ and $\omega_A$ at $E$ and $E'$, respectively, and $AB$ is tangent to $\omega$ and $\omega_A$ at $F$ and $F'$ respectively. Furthermore, let $P$ and $Q$ be the intersections of $BI$ with $EF$ and $CI$ with $EF$, respectively, and let $P'$ and $Q'$ be the intersections of $BI_A$ with $E'F'$ and $CI_A$ with $E'F'$, respectively. Given that the circumradius of $\vartriangle ABC$ is a, compute the maximum integer value of $BC$ such that the area $[P QP'Q']$ is less than or equal to $1$. [b]p21.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. Let $c$ be a positive integer such that $gcd(b, c) = 1$. From each ordered pair $(x, y)$ such that $x$ and $y$ are both integers, we draw two lines through that point in the $x-y$ plane, one with slope $\frac{b}{c}$ and one with slope $-\frac{c}{b}$ . Given that the number of intersections of these lines in $[0, 1)^2$ is a square number, what is the smallest possible value of $ c$? Note that $[0, 1)^2$ refers to all points $(x, y)$ such that $0 \le x < 1$ and $ 0 \le y < 1$.

Numbers $1,\frac12,\frac13,\ldots,\frac1{2001}$ are written on a blackboard. A student erases two numbers $x,y$ and writes down the number $x+y+xy$ instead. Determine the number that will be written on the board after $2000$ such operations.

How many integer solutions $ (x_0$, $ x_1$, $ x_2$, $ x_3$, $ x_4$, $ x_5$, $ x_6)$ does the equation \[ 2x_0^2\plus{}x_1^2\plus{}x_2^2\plus{}x_3^2\plus{}x_4^2\plus{}x_5^2\plus{}x_6^2\equal{}9\] have?

An $n$-tuple $(a_1, a_2 . . . , a_n)$ is [i]occasionally periodic[/i] if there exist a non-negative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+j+p}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, a_2 . . . , a_n)$ with elements from the set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, a_2, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$.

Given a rectangle $\mathcal{R}$ of area $100000 $, Pancho must completely cover the rectangle $\mathcal{R}$ with a finite number of rectangles with sides parallel to the sides of $\mathcal{R}$ . Next, Martín colors some rectangles of Pancho's cover red so that no two red rectangles have interior points in common. If the red area is greater than $0.00001$, Martin wins. Otherwise, Pancho wins. Prove that Pancho can cover to ensure victory,

Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$ b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?

Is it is possible to choose 24 distinct points in the space such that no three of them lie on the same line and choose 2019 distinct planes in a way that each plane passes through at least 3 of the chosen points and each triple belongs to one of the chosen planes?

Three schools each have $n$ students. Each student knows a total of $n + 1$ students at the other two schools. Show that there must be three students, one from each school, who know each other.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years. (a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time. Consider a period of $rb$ years in which the column is initially empty. Determine, in terms of $r$ and $b$, the number of blocks in the column at the end. (b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd. At time $t=0$, the column is initially empty. Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years. Prove that at time $t=rb$, the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$, where \[ S = \left\lfloor \frac{2r}{r+b} \right\rfloor + \left\lfloor \frac{4r}{r+b} \right\rfloor + ... + \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor . \] [i]Proposed by Sammy Luo[/i]

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]

A square of side $1$ is divided into rectangles . We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we choose any of the four sides) . Prove that the sum of the lengths of all the chosen sides is at least $1$ . (Folklore)

On a plane, $n$ straight lines are drawn. Two domains are called [i]adjacent [/i] if they border by a line segment. Prove that the domains into which the plane is divided by these lines can be painted two colors so that no two [i]adjacent [/i] domains are of the same color.

In a class, there are $n{}$ children of different heights. Denote by $A{}$ the number of ways to arrange them all in a row, numbered $1,2,\ldots,n$ from left to right, so that each person with an odd number is shorter than each of his neighbors. Let $B{}$ be the number of ways to organize $n-1$ badminton games between these children so that everyone plays at most two games with children shorter than himself and at most one game with children taller than himself (the order of the games is not important). Prove that $A = B$.