Find the smallest positive integer $n$ with the following property: After painting black exactly $n$ cells of a $7\times 7$ board there always exists a $2\times 2$ square with at least three black cells.

Given a set $L$ of lines in general position in the plane (no two lines in $L$ are parallel, and no three lines are concurrent) and another line $\ell$, show that the total number of edges of all faces in the corresponding arrangement, intersected by $\ell$, is at most $6|L|$. [i]Chazelle et al., Edelsbrunner et al.[/i]

Consider a $3\times7$ grid of squares. Each square may be coloured green or white. [list] (a) Is it possible to find a colouring so that no subrectangle has all four corner squares of the same colour? (b) Is it possible for a $4\times 6$ grid? [/list] [i]Subrectangles must have their corners at grid-points of the original diagram. The corner squares of a subrectangle must be different. The original diagram is a subrectangle of itself.[/i]

Batman, Robin, and The Joker are in three of the vertex cells in a square $2025 \times 2025$ board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker. If in each move all three can see where the others moved, who has a winning strategy, The Joker, or Batman and Robin? Explain the answer. [b]Comment.[/b] Batman and Robin decide their common strategy at the beginning. [img]https://i.imgur.com/PeLBQNt.png[/img]

Place in the cells the boards measuring: a) $2 \times 2$, b) $4 \times 4$, c) $2n \times 2n$, numbers $0$, $1$ and $-1$ so that in each case all the sums of numbers in rows and columns are different.

In the rectangle in the figure, we are going to write $12$ numbers from $1$ to $9$, so that the sum of the four numbers written in each line is the same and the sum of the three is also equal numbers in each column. Six numbers have already been written. Determine the sum of the numbers of each row and every column. [img]https://cdn.artofproblemsolving.com/attachments/7/f/3db9ded1e703c5392f258e1608a1800760d78c.png[/img]

There are $256$ players in a tennis tournament who are ranked from $1$ to $256$, with $1$ corresponding to the highest rank and $256$ corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$. In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner. Determine the expected value of the rank of the winner.

There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she flips tails, she moves the token one square to the left. At the end of the ten flips, the probability that the token finishes on a red square is a rational number of the form $\frac a b$. Given that $a + b = 2001$, determine the largest possible value for $n$.

Consider a partition of $\{1,2,\ldots,900\}$ into $30$ subsets $S_1,S_2,\ldots,S_{30}$ each with $30$ elements. In each $S_k$, we paint the fifth largest number blue. Is it possible that, for $k=1,2,\ldots,30$, the sum of the elements of $S_k$ exceeds the sum of the blue numbers?

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N \ge 3$ be the answer to Problem 21. A regular $N$-gon is inscribed in a circle of radius $1$. Let $D$ be the set of diagonals, where we include all sides as diagonals. Then, let $D'$ be the set of all unordered pairs of distinct diagonals in $D$. Compute the sum $$\sum_{\{d,d'\}\in D'} \ell (d)^2 \ell (d')^2,$$where $\ell (d)$ denotes the length of diagonal $d$. [b]p20.[/b] Let $N$ be the answer to Problem $19$, and let $M$ be the last digit of $N$. Let $\omega$ be a primitive $M$th root of unity, and define $P(x)$ such that$$P(x) = \prod^M_{k=1} (x - \omega^{i_k}),$$where the $i_k$ are chosen independently and uniformly at random from the range $\{0, 1, . . . ,M-1\}$. Compute $E \left[P\left(\sqrt{\rfloor \frac{1250}{N} \rfloor } \right)\right].$ [b]p21.[/b] Let $N$ be the answer to Problem $20$. Define the polynomial $f(x) = x^{34} +x^{33} +x^{32} +...+x+1$. Compute the number of primes $p < N$ such that there exists an integer $k$ with $f(k)$ divisible by $p$.

A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.

The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight.

A $9\times 1$ rectangle is divided into unit squares. A broken line, from the lower left to the upper right corner, goes through all $20$ vertices of the unit squares and consists of $19$ line segments. How many such lines are there?

A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line. (a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides? (b) Do there exist three such $7$-gons? (V Proizvolov)

Let $n \ge 3$ be an odd integer. Each cell is a $n \times n$ board painted in yellow or blue. Let's call the sequence of cells $S_1, S_2,...,S_m$ [i]path [/i] if they are all the same color and the cells $S_i$ and $S_j$ have one in common an edge if and only if $|i - j| = 1$. Suppose that all yellow cells form a path and all the blue cells form a path. Prove that one of the two paths begins or ends at the center of the board.

At least $3$ players take part in a tennis tournament. Each participant plays exactly one match against each other participant. After the tournament has ended, we find out that each player has won at least one match. (There are no ties in tennis). Show that in the tournament, there was at least one trio of players $A,B,C$ such that $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.

Let $A$ be an infinite set of real numbers containing at least one irrational number. Prove that for every natural number $n > 1$ there exists a subset $S$ of $A$ with n elements such that the sum of the elements of $S$ is an irrational number.

Hexagonal pieces numbered by positive integers are placed on the cells of a hexagonal board with side $n$. Two adjacent cells are left empty, and thanks to it some pieces can be moved. Two pieces with common sides exchanged places (see an example in the attachment 2). Prove that if $n \ge 3$ the second arrangement cannot be obtained from the first one by moving piece Note. Moving a piece a requires two adjacent empty cells. For instance, if they are on the right of a (attachment 1, left figure), a can be moved right till it touches an angle (attachment 1, middle figure), and then it can be moved upward right or downward right (attachment 1, right figure)

Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges. [hide=*]A tree is a connected graph with no cycles. A leaf is a vertex of degree 1[/hide] [i]Vincent Huang[/i]

A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If $x, u,$ and $v$ are three distinct vertices such that $x \to u$ and $x \to v$, then $u \to w$ and $v \to w$ for some vertex $w$. Suppose that $x \to u \to y \to\cdots \to z$ is a path of length $n$, that cannot be extended to the right (no arrow goes away from $z$). Prove that every path beginning at $x$ arrives after $n$ steps at $z.$

In the plane there are $2020$ points, some of which are black and the rest are green. For every black point, the following applies: [i]There are exactly two green points that represent the distance $2020$ from that black point. [/i] Find the smallest possible number of green dots. (Walther Janous)

Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.

Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.