The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut). Determine the largest positive integer $M$ such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least $M$. (Cells with coordinates $(x,y)$ and $(x',y')$ are considered to be neighbours if $x=x'$ and $y-y'\equiv\pm1\pmod{2011}$, or if $y=y'$ and $x-x'\equiv\pm1\pmod{2011}$.) [i](Romania) Dan Schwarz[/i]

You are cheating at a trivia contest. For each question, you can peek at each of the $n > 1$ other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth $0$ points. An incorrect guess is worth $-2$ points for other contestants, but only $-1$ point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by $2^{n - 1}$ points at any time, then you can surely win first place. [i]Linus Hamilton[/i]

Given $100$ points on the plane. Prove that you can cover them with a family of circles with the sum of their diameters less than $100$ and the distance between any two of the circles more than one.

The cheering team of Ubon Ratchathani University sits on the amphitheater that has $441$ seats arranged into a $21\times 21$ grid. Every seat is occupied by exactly one person, and each person has a blue sign and a yellow sign. Count the number of ways for each person to raise one sign so that each row and column has an odd number of people raising a blue sign.

Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.

Let $ n, k \in \mathbb{N}$ with $ 1 \leq k \leq \frac {n}{2} - 1.$ There are $ n$ points given on a circle. Arbitrarily we select $ nk + 1$ chords among the points on the circle. Prove that of these chords there are at least $ k + 1$ chords which pairwise do not have a point in common.

A rectangle $ABCD$ is given. On the side $AB$, n different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$ between $A$ and $D$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way? An example of a particular rectangle $ABCD$ is shown with a shaded one rectangle that may be formed in this way. [img]https://cdn.artofproblemsolving.com/attachments/e/4/f7a04300f0c846fb6418d12dc23f5c74b54242.png[/img]

Six musicians gathered at a chamber music festival . At each scheduled concert some of these musicians played while the others listened as members of the audience . What is the least number of such concerts which would need to be scheduled in order to enable each musician to listen , as a member of the audience, to all the other musicians? (Canadian origin)

Simultaneously from the same point of a circular route and in the same direction for two hours two bodies move evenly. The first body performs a complete rotation three minutes faster than the second body and exceeds it every $9$ minutes and $20$ seconds. Whenever the first body will overtake the other the second exactly at the starting point?

Given is natural number $n$. Sasha claims that for any $n$ rays in space, no two of which have a common point, he will be able to mark on these rays $k$ points lying on one sphere. What is the largest $k$ for which his statement is true?

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

A and B two people are throwing n fair coins.X and Y are the times they get heads. If throwing coins are mutually independent events, (1)When n=5, what is the possibility of X=Y? (2)When n=6, what is the possibility of X=Y+1?

Find the largest constant $c$, such that if there are $N$ discs in the plane such that every two of them intersect, then there must exist a point which lies in the common intersection of $cN + O(1)$ discs

[b]a)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n$? [b]b)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n^2$? [i]Proposed by Morteza Saghafian[/i]

Island Hopping Holidays offer short holidays to $64$ islands, labeled Island $i, 1 \le i \le 64$. A guest chooses any Island $a$ for the first night of the holiday, moves to Island $b$ for the second night, and finally moves to Island $c$ for the third night. Due to the limited number of boats, we must have $b \in T_a$ and $c \in T_b$, where the sets $T_i$ are chosen so that (a) each $T_i$ is non-empty, and $i \notin T_i$, (b) $\sum^{64}_{i=1} |T_i| = 128$, where $|T_i|$ is the number of elements of $T_i$. Exhibit a choice of sets $T_i$ giving at least $63\cdot 64 + 6 = 4038$ possible holidays. Note that c = a is allowed, and holiday choices $(a, b, c)$ and $(a',b',c')$ are considered distinct if $a \ne a'$ or $b \ne b'$ or $c \ne c'$.

South Adrican Magical Flights (SAMF) operates flights between South Adrican airports. If there is a flight from airport $A$ to airpost $B$, there will be also a flight from $B$ to $A$. The SAMF headquarters are located in Kimberley. Every airport that is served by Kimberley can be reached from Kimberley in precisely one way. This way of reaching Kimberley may involve stopping at other airports on the way. (For example, it may happen that you can get to Kimberley by flying from Durban to Bloemfontein and then from to Bloemfontein to Kimberley. In that case there is no other way to get from Durban to Kimberley. For example, there would be no direct Hight from Durban to Kimberley.) An airport (other than Kimberley) is called terminal if there are flights to (and from) precisely one other airport. Suppose that there are $t$ terminal airports. Due to budget cuts, SAMF decides to close down $k$ of the airports. It should still be possible to reach each of the remaining airports from Kimberley. Let $C$ be the number of choices for the $k$ destinations that are discontinued. Prove that $$\frac{t!}{k!(t-k)} \le C \le \frac{(t+k-1)!}{k!(t-1)!} .$$

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We say that $ n$ squares in a $ n\times n$ board are scattered if no two of them are in the same row or column.In every square of this board is witten a natural number so that the sum of numbrs in $ n$ scattered squares is always the same and no row or no column contains two equal numbers .It turned out that the numbers on the main diagonal are arranged in the increasing order ,and that their product is the smallest among all products of $ n$ scattered numbers .Prove that scattered numbers with the greatest product are exactly those on the other diagonal.

There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he wins the fight; otherwise, the stronger boxer wins. It is known that there are three boxers who obtained (strictly) more wins than the strongest three boxers. What is the minimum number of plotters ? [i]Proposed by N. Kalinin[/i]

Twelve places have been arranged at a round table for members of the Jury, with a name tag at each place . Professor K. being absent-minded instead of occupying his place, sits down at the next place (clockwise) . Each of the other Jury members in turn either occupies the place assigned to this member or, if it has been already occupied, sits down at the first free place in the clockwise order. The resulting seating arrangement depends on the order in which the Jury members come to the table. How many different seating arrangements of this kind are possible? (A Shapovalov)

In fields of a $19 \times 19$ table are written integers so that any two lying on neighboring fields differ at most by $2$ (two fields are neighboring if they share a side). Find the greatest possible number of mutually different integers in such a table.

We wish to write $n$ distinct real numbers $(n\geq3)$ on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of $n$ such that this is possible.

Is it possible to put distinct positive integers less than $1991$ in the cells of a $9\times 9$ table so that the products of all the numbers in every column and every row are equal to each other? (N.B. Vasiliev, Moscow)