Three schools have $200$ students each. Every student has at least one friend in each school (if the student $a$ is a friend of the student $b$ then $b$ is a friend of $a$). It is known that there exists a set $E$ of $300$ students (among the $600$) such that for any school $S$ and any two students $x,y\in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different. Show that one can find a student in each school such that they are friends with each other.

Given a positive integer $ n\geq 2$, consider a set of $ n$ islands arranged in a circle. Between every two neigboring islands two bridges are built as shown in the figure. Starting at the island $ X_1$, in how many ways one can one can cross the $ 2n$ bridges so that no bridge is used more than once?

In each side of a regular polygon with $n$ sides, we choose a point different from the vertices and we obtain a new polygon of $n$ sides. For which values of $n$ can we obtain a polygon such that the internal angles are all equal but the polygon isn't regular?

How many ways are there of representing a positive integer $n$ as the sum of three positive integers? Representations which differ only in the order of the summands are considered to be distinct.

A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ [i]Dan Schwarz, Romania[/i]

Consider the integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$. [b](a)[/b] Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$, where $j$ is not a function of $x$, is $Z(j)=e^{j^{2}/4a} Z(0)$. [b](b)[/b] Show that $$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n},$$ where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times\cdots\times3\times 1$. [b](c)[/b] What is the number of ways to form $n$ pairs from $2n$ distinct objects? Interpret the previous part of the problem in term of this answer.

Ali has $100$ cards with numbers $1,2,\ldots,100$. Ali and Amin play a game together. In each step, first Ali chooses a card from the remaining cards and Amin decides to pick that card for himself or throw it away. In the case that he picks the card, he can't pick the next card chosen by Amin, and he has to throw it away. This action repeats until when there is no remaining card for Ali. Amin wants to pick cards in a way that the sum of the number of his cards is maximized and Ali wants to choose cards in a way that the sum of the number of Amin's cards is minimized. Find the most value of $k$ such that Amin can play in a way that is sure the sum of the number of his cards will be at least equal to $k$.

Let $\overline{abcd}$ be $4$ digit number, such that we can do transformations on it. If some two neighboring digits are different than $0$, then we can decrease both digits by $1$ (we can transform $9870$ to $8770$ or $9760$). If some two neighboring digits are different than $9$, then we can increase both digits by $1$ (we can transform $9870$ to $9980$ or $9881$). Can we transform number $1220$ to: $a)$ $2012$ $b)$ $2021$

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Let $P_{1}, P_{2}, \ldots, P_{2021}$ be 2021 points in the quarter plane $\{(x, y): x \geq 0, y \geq 0\}$. The centroid of these 2021 points lies at the point $(1,1)$. Show that there are two distinct points $P_{i}, P_{j}$ such that the distance from $P_{i}$ to $P_{j}$ is no more than $\sqrt{2} / 20$.

Two sequences $b_i$, $c_i$, $0 \le i \le 100$ contain positive integers, except $c_0=0$ and $b_{100}=0$. Some towns in Graphland are connected with roads, and each road connects exactly two towns and is precisely $1$ km long. Towns, which are connected by a road or a sequence of roads, are called [i]neighbours[/i]. The length of the shortest path between two towns $X$ and $Y$ is denoted as [i]distance[/i]. It is known that the greatest [i]distance[/i] between two towns in Graphland is $100$ km. Also the following property holds for every pair $X$ and $Y$ of towns (not necessarily distinct): if the [i]distance[/i] between $X$ and $Y$ is exactly $k$ km, then $Y$ has exactly $b_k$ [i]neighbours[/i] that are at the [i]distance[/i] $k+1$ from $X$, and exactly $c_k$ [i]neighbours[/i] that are at the [i]distance[/i] $k-1$ from $X$. Prove that $$\frac{b_0b_1 \cdot \cdot \cdot b_{99}}{c_1c_2 \cdot \cdot \cdot c_{100}}$$ is a positive integer.

[b]p1.[/b] Find the smallest positive integer which is $1$ more than multiple of $3$, $2$ more than a multiple of $4$, and $4$ more than a multiple of $7$. [b]p2.[/b] Let $p = 4$, and let $a =\sqrt1$, $b =\sqrt2$, $c =\sqrt3$, $...$. Compute the value of $(p-a)(p-b) ... (p-z)$. [b]p3.[/b] There are $6$ points on the circumference of a circle. How many convex polygons are there having vertices on these points? [b]p4.[/b] David and I each have a sheet of computer paper, mine evenly spaced by $19$ parallel lines into $20$ sections, and his evenly spaced by $29$ parallel lines into $30$ sections. If our two sheets are overlayed, how many pairs of lines are perfectly incident? [b]p5.[/b] A pyramid is created by stacking equilateral triangles of balls, each layer having one fewer ball per side than the triangle immediately beneath it. How many balls are used if the pyramid’s base has $5$ balls to a side? [b]p6.[/b] Call a positive integer $n$ good if it has $3$ digits which add to $4$ and if it can be written in the form $n = k^2$, where $k$ is also a positive integer. Compute the average of all good numbers. [b]p7.[/b] John’s birthday cake is a scrumptious cylinder of radius $6$ inches and height $3$ inches. If his friends cut the cake into $8$ equal sectors, what is the total surface area of a piece of birthday cake? [b]p8.[/b] Evaluate $\sum^{10}_{i=1}\sum^{10}_{j=1} ij$. [b]p9.[/b] If three numbers $a$, $b$, and $c$ are randomly selected from the interval $[-2, 2]$, what is the probability that $a^2 + b^2 + c^2 \ge 4$? [b]p10.[/b] Evaluate $\sum^{\infty}_{x=2} \frac{2}{x^2 - 1}.$ [b]p11.[/b] Consider $4x^2 - kx - 1 = 0$. If the roots of this polynomial are $\sin \theta$ and $\cos \theta$, compute $|k|$. [b]p12.[/b] Given that $65537 = 2^{16} + 1$ is a prime number, compute the number of primes of the form $2^n + 1$ (for $n \ge 0$) between $1$ and $10^6$. [b]p13.[/b] Compute $\sin^{-1}(36/85) + \cos^{-1}(4/5) + \cos^{-1}(15/17).$ [b]p14.[/b] Find the number of integers $n$, $1\le n \le 2003$, such that $n^{2003} - 1$ is a multiple of $10$. [b]p15.[/b] Find the number of integers $n,$ $1 \le n \le 120$, such that $n^2$ leaves remainder $1$ when divided by $120$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Maryam labels each vertex of a tetrahedron with the sum of the lengths of the three edges meeting at that vertex. She then observes that the labels at the four vertices of the tetrahedron are all equal. For each vertex of the tetrahedron, prove that the lengths of the three edges meeting at that vertex are the three side lengths of a triangle.

$n$ children want to divide $m$ identical pieces of chocolate into equal amounts, each piece being broken not more than once. (a) For what $n$ is it possible, if $m = 9$? (b) For what $n$ and $m$ is it possible? (Y. Tschekanov, Moscow)

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

Let $C$ be the set of points $(x,y)$ with integer coordinates in the plane where $1\leq x\leq 900$ and $1\leq y\leq 1000$. A polygon $P$ with vertices in $C$ is called [i]emerald[/i] if $P$ has exactly zero or two vertices in each row and each column and all the internal angles of $P$ are $90^\circ$ or $270^\circ$. Find the greatest value of $k$ such that we can color $k$ points in $C$ such that any subset of these $k$ points is not the set of vertices of an [i]emerald[/i] polygon. [img]https://cdn.discordapp.com/attachments/954427908359876608/1299737432010395678/image.png?ex=671e4a4f&is=671cf8cf&hm=ce008541975226a0e9ea53a93592a7469d8569baca945c1c207d4a722126bb60&[/img] On the left, an example of an emerald polygon; on the right, an example of a non-emerald polygon.

[b]p1.[/b] To convert between Fahrenheit, $F$, and Celsius, $C$, the formula is $F = \frac95 C + 32$. Jennifer, having no time to be this precise, instead approximates the temperature of Fahrenheit, $\widehat F$, as $\widehat F = 2C + 30$. There is a range of temperatures $C_1 \le C \le C_2$ such that for any $C$ in this range, $| \widehat F - F| \le 5$. Compute the ordered pair $(C_1,C_2)$. [b]p2.[/b] Compute integer $x$ such that $x^{23} = 27368747340080916343$. [b]p3.[/b] The number of ways to flip $n$ fair coins such that there are no three heads in a row can be expressed with the recurrence relation $$ S(n + 1) = a_0 S(n) + a_1 S(n - 1) + ... + a_k S(n - k) $$ for sufficiently large $n$ and $k$ where $S(n)$ is the number of valid sequences of length $n$. What is $\sum^k_{n=0}|a_n|$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Thomas and Nils are playing a game. They have a number of cards, numbered $1, 2, 3$, et cetera. At the start, all cards are lying face up on the table. They take alternate turns. The person whose turn it is, chooses a card that is still lying on the table and decides to either keep the card himself or to give it to the other player. When all cards are gone, each of them calculates the sum of the numbers on his own cards. If the difference between these two outcomes is divisible by $3$, then Thomas wins. If not, then Nils wins. (a) Suppose they are playing with $2018$ cards (numbered from $1$ to $2018$) and that Thomas starts. Prove that Nils can play in such a way that he will win the game with certainty. (b) Suppose they are playing with $2020 $cards (numbered from $1$ to $2020$) and that Nils starts. Which of the two players can play in such a way that he wins with certainty?

We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.

In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the $n$ -th switch the $n$ -th bulb will light up. There is a group of men who are operating the switches according to the following rule: they go in one by one and starts flipping the switches starting from the first switch until he has to turn on a bulb; as soon as he turns a bulb on, he leaves the room. For example the first person goes in, turns the first switch on and leaves. Then the second man goes in, seeing that the first switch on turns it off and then lights the second bulb. Then the third person goes in, finds the first switch off and turns it on and leaves the room. Then the fourth person enters and switches off the first and second bulbs and switches on the third. The process continues in this way. Finally we find out that first 10 bulbs are off and the 11 -th bulb is on. Then how many people were involved in the entire process?

We say that a set of points $M$ in the plane is convex if for any two points $A, B \in M$, all the points from the segment $(AB)$ also belong to $M$. Let $n \ge 2$ be an integer and let $F$ be a family of $n$ disjoint convex sets in the plane. Prove that there exists a line $\ell$ in the plane, a set $M \in F$ and a subset $S \subset F$ with at least $\lceil \frac{n}{12} \rceil $ elements such that $M$ is contained in one closed half-plane determined by $\ell$, and all the sets $N \in S$ are contained in the complementary closed half-plane determined by $\ell$.

You have an infinite stack of T-shaped tetrominoes (composed of four squares of side length 1), and an n × n board. You are allowed to place some tetrominoes on the board, possibly rotated, as long as no two tetrominoes overlap and no tetrominoes extend off the board. For which values of n can you cover the entire board?

A table has $3 000 000$ rows and $100$ columns, divided into unit squares. Each row contains the numbers from $1$ to $100$, each exactly once, and no two rows are the same. Above each column, the number of distinct entries in that column is written in red. Find the smallest possible sum of the red numbers.

A convex pentagon $ABCDE$ satisfies that the sidelengths and $AC,AD\leq \sqrt 3.$ Let us choose $2011$ distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least $403$ points out of the $2011$ given points. {Edited} {I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}