Given that: $\tfrac{6}{11}-\tfrac{10}{19}=\tfrac{9}{19}-\tfrac{n}{11}$, find $n$.

Determine all integers $n\geqslant 3$ such that there exist $n{}$ pairwise distinct real numbers $a_1,\ldots,a_n$ for which the sums $a_i+a_j$ over all $1\leqslant i<j\leqslant n$ form an arithmetic progression.

Give an example of ten different noncoplanar points $ P_1,\ldots ,P_5,Q_1,\ldots ,Q_5$ in $ 3$-space such that connecting each $ P_i$ to each $ Q_j$ by a rigid rod results in a rigid system. [i]L. Lovasz[/i]

Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy: $\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.

In a math competition, $14$ schools participate, each sending $14$ students. The students are separated into $14$ groups of $14$ so that no two students from the same school are in the same group. The tournament organizers noted that, from the competitors, exactly $15$ have participated in the competition before. The organizers want to select two representatives, with the conditions that they must be former participants, must come from different schools, and must also be in different groups. It turns out that there are $ n$ ways to do this. What is the minimum possible value of $n$?

Let $X$ be normally distributed with mean $\mu$ and variance $\sigma^2>0$. What is the variance of $e^X$?

One of the following numbers is not divisible by any prime number less than 10. Which is it? (A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $

Define a regular $n$-pointed star to be a union of $n$ lines segments $P_1P_2, P_2P_3, ..., P_nP_1$ such that $\bullet$ the points $P_1,P_2,...,P_n$ are coplanar and no three of them are collinear, $\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint, $\bullet$ all of the angles at $P_1, P_2,..., P_n$ are congruent , $\bullet$ all of the $n$ line segments $P_1P_2, P_2P_3, ..., P_nP_1$ are congruent, and $\bullet$ the path $P_1P_2...P_nP_1$ turns counterclockwise at an angle less than $180^o$ at each vertex. There are no regular $3$-pointed, $4$-pointed, or $6$-pointed stars. All regular $5$-pointed star are similar, but there are two non-similar regular $7$-pointed stars. Find all possible values of $n$ such that there are exactly $29$ non-similar regular $n$-pointed stars.

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.

Let $k\geq 2$ be an integer. We want to determine the weight of $n$ balls. One try consists of choosing two balls, and we are given the sum of the weights of the two chosen balls. We know that at most $k$ of the answers can be wrong. Let $f_k(n)$ denote the smallest number for which it is true that we can always find the weights of the balls with $f_k(n)$ tries (the tries don't have to be decided in advance). Prove that there exist numbers $a_k$ and $b_k$ for which $|f_k(n)-a_kn|\leq b_k$ holds. [i]Proposed by Surányi László, Budapest and Bálint Virág, Toronto[/i]

Find the number of ordered triples of positive integers $\left(a,b,c\right)$ such that $a\times b\times c=2008^2$.

In a certain city there are $n$ straight streets, such that every two streets intersect, and no three streets pass through the same intersection. The City Council wants to organize the city by designating the main and the side street on every intersection. Prove that this can be done in such way that if one goes along one of the streets, from its beginning to its end, the intersections where this street is the main street, and the ones where it is not, will apear in alternating order. [i]Proposed by Serbia[/i]

Snow White has, in her huge basket, $2021$ apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seven dwarfs. Therefore, she decided to use them to find out which apple is poisonous. To this end, at the beginning of each day, Snow White prepares some apple salads (each salad is a mixture of pieces of some apples chosen by her), and forces some of the dwarfs (possibly all) to eat a salad each. At the end of the day, she notes who died and who survived, and the next day she again prepares some apple salads, forcing some of the surviving dwarves (possibly all) to eat a salad each. And she continues to do this, day after day, until she discovers the poisoned apple or until all the dwarves die. (a) Prove that there is a strategy in which Snow White manages to discover the poisoned apple after a few days. (b) What is the minimum number of days Snow White needs to find the poisoned apple, no matter how lucky she is?

Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding $5000$. Then she fixes $20$ distinct positive integers $a_1, a_2, \cdots, a_{20}$ such that, for each $k = 1,2,\cdots,20$, the numbers $N$ and $a_k$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of positive integers not exceeding $20$, and she tells him back the set $\{a_k : k \in S\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of? [i]Sergey Kudrya, Russia[/i]

It is known that a $n$-vertex contains within itself a polyhedron $M$ with a center of symmetry at some point $Q$ and is itself contained in a polyhedron homothetic to $M$ with a homothety center at a point $Q$ and coefficient $k$. Find the smallest value of $k$ if a) $n = 4$, b) $n = 5$.

In Thailand there are $n$ cities. Each pair of cities is connected by a one-way street which can be borrowed, depending on its type, only by bike or by car. Show that there is a city from which you can reach any other city, either by bike or by car. [i]Remark : It is not necessary to use the same means of transport for each city[/i]

A subset $S$ of size $n$ of a plane consisting of points with both coordinates integer is given, where $n$ is an odd number. The injective function $f\colon S\rightarrow S$ satisfies the following: for each pair of points $A, B\in S$, the distance between points $f(A)$ and $f(B)$ is not smaller than the distance between points $A$ and $B$. Prove there exists a point $X$ such that $f(X)=X$.

A closed, non-self-intersecting broken line $L$ is drawn over a $(2n+1) \times (2n+1)$ chessboard in such a way that the set of L's vertices coincides with the set of the vertices of the board’s squares and every edge in $L$ is a side of some board square. All board squares lying in the interior of $L$ are coloured in red. Prove that the number of neighbouring pairs of red squares in every row of the board is even.

Some of the values produced by two functions, $f(x)$ and $g(x)$, are shown below. Find $f(g(3))$ \begin{tabular}{c||c|c|c|c|c} $x$ & 1 & 3 & 5 & 7 & 9 \\ \hline\hline $f(x)$ & 3 & 7 & 9 & 13 & 17 \\ \hline $g(x)$ & 54 & 9 & 25 & 19 & 44 \end{tabular} $\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }13\qquad\textbf{(E) }17$

Let $ABC$ be an acute triangle with orthocenter $H$, and let $P$ be a point on the nine-point circle of $ABC$. Lines $BH, CH$ meet the opposite sides $AC, AB$ at $E, F$, respectively. Suppose that the circumcircles $(EHP), (FHP)$ intersect lines $CH, BH$ a second time at $Q,R$, respectively. Show that as $P$ varies along the nine-point circle of $ABC$, the line $QR$ passes through a fixed point. [i]Proposed by Brandon Wang[/i]

Suppose $I$ and $I_A$ are the incenter and the $A$-excenter of triangle $ABC$, respectively. Let $M$ be the midpoint of arc $BAC$ on the circumcircle, and $D$ be the foot of the perpendicular from $I_A$ to $BC$. The line $MI$ intersects the circumcircle again at $T$ . For any point $X$ on the circumcircle of triangle $ABC$, let $XT$ intersect $BC$ at $Y$ . Prove that $A, D, X, Y$ are concyclic.

The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\] Show that $ a_{2008} < 5$.

Let $n$ be a positive integer. Find the number of permutations $\sigma$ of the set $\{1, 2, ..., n\}$ such that $\sigma(j) \geq j$ holds for exactly two values of $j$.

Find the locus of all numbers $z\in\mathbb C$ in complex plane satisfying $$z+\bar z=a\cdot|z|,$$ where $a\in\mathbb R$ is given.