2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.

Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.

Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.

The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm. [img]https://cdn.artofproblemsolving.com/attachments/4/2/cda98a00f8586132fe519855df123534516b50.png[/img] a) What is the height of the liquid when it lies as shown in figure $2$? b) What is the height of the liquid when it lies as shown in figure$ 3$?

In a given community of people, each person has at least two friends within the community. Whenever some people from this community sit on a round table such that each adjacent pair of people are friends, it happens that no non-adjacent pair of people are friends. Prove that there exist two people in this community such that each has exactly two friends and they have at least one common friend.

Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]

Which of the following answer choices is equivalent to $\sqrt{a^3b^2c}$? $\textbf{(A) }ab\sqrt{ac}\qquad\textbf{(B) }bc\sqrt{ac}\qquad\textbf{(C) }b\sqrt{ac}\qquad\textbf{(D) }abc\sqrt{ab}\qquad\textbf{(E) }a\sqrt{bc}$

$100$ Queens are placed on a $100 \times 100$ chessboard so that no two attack each other. Prove that each of four $50 \times 50$ corners of the board contains at least one Queen.

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

A function $f: \N\rightarrow\N$ is circular if for every $p\in\N$ there exists $n\in\N,\ n\leq{p}$ such that $f^n(p)=p$ ($f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\in\N$ $f^i(p)\neq{p}$ for every $i=1,2,\dots,\lfloor{kp}\rfloor$. Determine the maximum repulsion degree can have a circular function. [b]Note:[/b] Here $\lfloor{x}\rfloor$ is the integer part of $x$.

Consider a $1$ by $2$ by $3$ rectangular prism. Find the length of the shortest path between opposite corners $A$ and $B$ that does not leave the surface of the prism. [img]https://cdn.artofproblemsolving.com/attachments/7/1/899b79b0f938e63fe7b093cecff63ed1255acb.png[/img]

Find all pairs of integers $(x, y)$ such that $x^2 + 5y^2 = 2021y$.

Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.

Given an integer $k\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\cdots x_k!$ for all positive integers $x_1,x_2,\cdots x_k$. $Albania$

Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$. Let $\omega$ be the circle that touches the sides $AB$, $BC$, and $AD$. A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$. Determine the ratio $\frac{AK}{KD}$. [i]Proposed by Giorgi Arabidze, Georgia[/i]

A function $f(x)$ defined for $x\ge 0$ satisfies the following conditions: i. for $x,y\ge 0$, $f(x)f(y)\le x^2f(y/2)+y^2f(x/2)$; ii. there exists a constant $M$($M>0$), such that $|f(x)|\le M$ when $0\le x\le 1$. Prove that $f(x)\le x^2$.

Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function with the property that $ (f\circ f) (x) =[x], $ for any real number $ x. $ Show that there exist two distinct real numbers $ a,b $ so that $ |f(a)-f(b)|\ge |a-b|. $ $ [] $ denotes the integer part.

Find all three digit positive integers that have distinct digits and after their greatest digit is switched to $1$ become multiples of $30$.

[b]Problem 4[/b] : A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent line to both circles touches the circle with $AC$ as diameter at $P \neq C$ and the circle with $CB$ as diameter at $Q \neq C.$ Prove that lines $AP, BQ$ and the common tangent line to both circles at $C$ all meet at a single point which lies on the circle with $AB$ as diameter.

Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$. The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there. Find all $n$ for which he can select the guests in such an order to serve all the guests.

A bank has one million clients, one of whom is Inspector Gadget. Each client has a unique PIN number consisting of six digits. Dr. Claw has a list of all the clients. He is able to break into the account of any client, choose any $n$ digits of the PIN number and copy them. The n digits he copies from different clients need not be in the same $n$ positions. He can break into the account of each client, but only once. What is the smallest value of $n$ which allows Dr.Claw to determine the complete PIN number of Inspector Gadget?

It is given that $2^{333}$ is a $101$-digit number whose first digit is $1$. How many of the numbers $2^k$, $1 \le k \le 332$, have first digit $4$?

Which of the following statements are true? (a) If a polygon can be divided into two congruent polygons by a broken line segment, it can be divided into two congruent polygons by a straight line segment. (b) If a convex polygon can be divided into two congruent polygons by a broken line segment, it can be so divided by a straight line segment. (c) If a convex polygon can be divided into two polygons by a broken line segment, one of which can be mapped onto the other by a combination of rotations and translations, it can be so divided by a straight line segment. (S Markelov,)

Let $ABCD$ be a cyclic quadrilateral in which internal angle bisectors $\angle ABC$ and $\angle ADC$ intersect on diagonal $AC$. Let $M$ be the midpoint of $AC$. Line parallel to $BC$ which passes through $D$ cuts $BM$ at $E$ and circle $ABCD$ in $F$ ($F \neq D$ ). Prove that $BCEF$ is parallelogram [i]Proposed by Steve Dinh[/i]