There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$. Define $S$ as the sum of the $interests-disparity$ of all the couples, $\binom{32}{2}\left ( =\: 496 \right )$ ones in total. Determine the minimal possible value of $S$.

Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?

A square of side $N=n^2+1$, $n\in \mathbb{N}^*$, is partitioned in unit squares (of side $1$), along $N$ rows and $N$ columns. The $N^2$ unit squares are colored using $N$ colors, $N$ squares with each color. Prove that for any coloring there exists a row or a column containing unit squares of at least $n+1$ colors.

Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids". Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

Alice and Bob are playing a game, starting with a binary string$ b$ of length $2022$. In each step, the rightmost digit of the string is deleted. If the deleted digit was $1$, Alice gets to choose which digit she wants to append on the left. Otherwise, Bob gets to choose the digit to append on the left of the string. Alice would like to turn the string $b$ into the all-zero string $\underbrace{00 . . . 0}_{2022}$, in the least number of steps possible, while Bob would like to maximize the number of steps necessary, or prevent Alice from doing this at all. a) Is there a string $b$ for which Bob can prevent Alice in her goal, if both players play optimally? b) If the answer to part a is yes, find all such strings $b$. If the answer is no, find the maximal game time and find the set of strings $b$ for which the game time is maximal.

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]

In a Cartesian plane, if both horizontal coordinate and vertical coordinate of a point are rational numbers, we call the point [i]rational point[/i]. Otherwise, we call it [i]irrational point[/i]. Consider an arbitrary regular pentagon on the Cartesian plane. Please compare the number of rational point and the number of irrational point among the five vertices of the pentagon. Prove your conclusion.

Every unit square of a $n \times n$ board is colored either red or blue so that among all 2 $\times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different). a) Find the least possible value of $n$. b) For the least possible value of $n$ find the least possible number of red unit squares

The company employs 50,000 people. For each of them, the sum of the number of his immediate superiors and his immediate subordinates is equal to 7. On Monday, each employee of the enterprise issues an order and gives a copy of this order to each of his direct subordinates (if there are any). Further, every day an employee takes all the basics he received on the previous day and either distributes them copies to all your direct subordinates, or, if any, he is not there, he carries out orders himself. It turned out that on Friday no papers were transferred to the institution. Prove that the enterprise has at least 97 bosses over whom there are no bosses.

There are $n\geq 2$ people at a meeting. Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting. (It is assumed that if $A$ is a friend of $B,$ then $B$ is a friend of $A;$ moreover, nobody is his own friend.)

Let $n,k$, $1\le k\le n$ be fixed integers. Alice has $n$ cards in a row, where the card has position $i$ has the label $i+k$ (or $i+k-n$ if $i+k>n$). Alice starts by colouring each card either red or blue. Afterwards, she is allowed to make several moves, where each move consists of choosing two cards of different colours and swapping them. Find the minimum number of moves she has to make (given that she chooses the colouring optimally) to put the cards in order (i.e. card $i$ is at position $i$). NOTE: edited from original phrasing, which was ambiguous.

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS" $A,\ B,\ C,\ H,\ F,\ N$. For example $AFHNNNHAFFC$ is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step $NA\rightarrow N$ the protein $BANANA$ will cahnge to $BANNA$("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the set of steps: $\\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null}$ Protein $ABBAABA$ will change like this: $\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB$ You see after finite steps this protein will finish it steps. Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous? a) $NO\longrightarrow OONN$ b) $\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.$ c) Design a set of allowed steps that change $\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}$ d) Design a set of allowed steps that change $\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}$ You see from $c$ and $d$ that we acn calculate the functions $F(n)=2^{n}$ and $G(M,N)=mn$ with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)

[b]p1.[/b] Arrange the whole numbers $1$ through $15$ in a row so that the sum of any two adjacent numbers is a perfect square. In how many ways this can be done? [b]p2.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 - q^2$ is divisible by $24$. [b]p3.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started ”None of you has $8$ legs. Only I have $8$ legs!” Which polyleg has exactly $8$ legs? [b][b]p4.[/b][/b] There is a small puncture (a point) in the wall (as shown in the figure below to the right). The housekeeper has a small flag of the following form (see the figure left). Show on the figure all the points of the wall where you can hammer in a nail such that if you hang the flag it will close up the puncture. [img]https://cdn.artofproblemsolving.com/attachments/a/f/8bb55a3fdfb0aff8e62bc4cf20a2d3436f5d90.png[/img] [b]p5.[/b] Assume $ a, b, c$ are odd integers. Show that the quadratic equation $ax^2 + bx + c = 0$ has no rational solutions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Professor Moriarty has designed a “prime-testing trail.” The trail has $2002$ stations, labeled $1,... , 2002$. Each station is colored either red or green, and contains a table which indicates, for each of the digits $0, ..., 9$, another station number. A student is given a positive integer $n$, and then walks along the trail, starting at station $1$. The student reads the first (leftmost) digit of $n,$ and looks this digit up in station $1$’s table to get a new station location. The student then walks to this new station, reads the second digit of $n$ and looks it up in this station’s table to get yet another station location, and so on, until the last (rightmost) digit of $n$ has been read and looked up, sending the student to his or her final station. Here is an example that shows possible values for some of the tables. Suppose that $n = 19$: [img]https://cdn.artofproblemsolving.com/attachments/f/3/db47f6761ca1f350e39d53407a1250c92c4b05.png[/img] Using these tables, station $1$, digit $1$ leads to station $29$m station $29$, digit $9$ leads to station $1429$, and station $1429$ is green. Professor Moriarty claims that for any positive integer $n$, the final station (in the example, $1429$) will be green if and only if $n$ is prime. Is this possible?

Some number of coins is firstly separated into 200 groups and then to 300 groups. One coin is [i]special[/i], if on the second grouping it is in a group that has less coins than the previous one, in the first grouping, that it was in. Find the least amount of [i]special[/i] coins we can have.

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

Let $S=\{ 1,2,3,\dots,26 \}.$ Find the number of ways in which $S$ can be partitioned into thirteen subsets such that the following is satisfied: [list] [*]each subset contains two elements of $S,$ and [*]the positive difference between the elements of a subset is $1$ or $13.$ [/list]

Let $n\geq 2$ be an integer. Consider $2n$ points around a circle. Each vertex has been tagged with one integer from $1$ to $n$, inclusive, and each one of these integers has been used exactly two times. Isabel divides the points into $n$ pairs, and draws the segments joining them, with the condition that the segments do not intersect. Then, she assigns to each segment the greatest integer between its endpoints. a) Show that, no matter how the points have been tagged, Isabel can always choose the pairs in such a way that she uses exactly $\lceil n/2\rceil$ numbers to tag the segments. b) Can the points be tagged in such a way that, no matter how Isabel divides the points into pairs, she always uses exactly $\lceil n/2\rceil$ numbers to tag the segments? Note. For each real number $x$, $\lceil x\rceil$ denotes the least integer greater than or equal to $x$. For example, $\lceil 3.6\rceil=4$ and $\lceil 2\rceil=2$. [i]Proposed by Victor Domínguez[/i]

$D$ is a closed disk radius $R$. Show that among any $8$ points of $D$ one can always find two whose distance apart is less than $R$.

There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property: We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

Let $n$ be an odd positive integer. Given are $n$ balls - black and white, placed on a circle. For a integer $1\leq k \leq n-1$, call $f(k)$ the number of balls, such that after shifting them with $k$ positions clockwise, their color doesn't change. a) Prove that for all $n$, there is a $k$ with $f(k) \geq \frac{n-1}{2}$. b) Prove that there are infinitely many $n$ (and corresponding colorings for them) such that $f(k)\leq \frac{n-1}{2}$ for all $k$.

Any of $6$ gossips has her own news. From time to time one of them makes a telephone call to some other gossip and they discuss fill the news they know. What the minimum number of the calls is necessary so as (for) all of them to know all the news?

The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.