A [i]$n$-trønder walk[/i] is a walk starting at $(0, 0)$, ending at $(2n, 0)$ with no self intersection and not leaving the first quadrant, where every step is one of the vectors $(1, 1)$, $(1, -1)$ or $(-1, 1)$. Find the number of $n$-trønder walks.

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

Let $ABC$ be a triangle with $AB=AC$. Let $D$ be the midpoint of $BC$, $M$ the midpoint of $AD$ and $N$ the foot of the perpendicular from $D$ to $BM$. Prove that $\angle ANC = 90^\circ$.

When the IMO is over and students want to relax, they all do the same thing: download movies from the internet. There is a positive number of rooms with internet routers at the hotel, and each student wants to download a positive number of bits. The load of a room is defined as the total number of bits to be downloaded from that room. Nobody likes slow internet, and in particular each student has a displeasure equal to the product of her number of bits and the load of her room. The misery of the group is defined as the sum of the students’ displeasures. Right after the contest, students gather in the hotel lobby to decide who goes to which room. After much discussion they reach a balanced configuration: one for which no student can decrease her displeasure by unilaterally moving to another room. The misery of the group is computed to be $M_1$, and right when they seemed satisfied, Gugu arrived with a serendipitous smile and proposed another configuration that achieved misery $M_2$. What is the maximum value of $M_1/M_2$ taken over all inputs to this problem? [i]Proposed by Victor Reis (proglote), Brazil.[/i]

Given an $n \times n$ ($n \ge 3$) square table with one of the following unit vectors $\uparrow, \downarrow, \leftarrow, \rightarrow$ in any its cell (the vectors are parallel to the sides and the middles of them coincide with the centers of the cells). Per move a beetle creeps from one cell to another in accordance with the vector’s direction. If the beetle starts from any cell, then it comes back to this cell after some number of moves. The vectors are directed so that they do not allow the beetle to leave the table. Is it possible that the sum of all vectors at any row (except for the first one and the last one) is equal to the vector that is parallel to this row, and the sum of all vectors at any column (except for the first one and the last one) is equal to the vector that is parallel to this column ? (D. Dudko)

Assume that the bisecting plane of the dihedral angle at edge $AB$ of the tetrahedron $ABCD$ meets the edge $CD$ at point $E$. Denote by $S_1, S_2, S_3$, respectively the areas of the triangles $ABC, ABE$, and $ABD$. Prove that no tetrahedron exists for which $S_1, S_2, S_3$ (in this order) form an arithmetic or geometric progression.

Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$. a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic. b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.

Let $ABC$ be an arbitrary triangle. Let $E,E$ be two points on $AB,AC$ respectively such that their distance to the midpoint of $BC$ is equal. Let $P$ be the second intersection of the triangles $ABC,AEF$ circumcircles . The tangents from $E,F$ to the circumcircle of $AEF$ intersect each other at $K$. Prove that : $\angle KPA = 90$

Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.

Find all vectors of $n$ real numbers $(x_1,x_2,\ldots,x_n)$ such that \[ \left\{ \begin{array}{ccc} x_1 & = & \dfrac 1{x_2} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n } \\ x_2 & = & \dfrac 1{x_1} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n} \\ \ & \cdots & \ \\ x_n & = & \dfrac 1{x_1} + \dfrac 1{x_2} + \cdots + \dfrac 1{x_{n-1}} \end{array} \right. \] [i]Mircea Becheanu[/i]

We are given a lattice and two pebbles $A$ and $B$ that are placed at two lattice points. At each step we are allowed to relocate one of the pebbles to another lattice point with the condition that the distance between pebbles is preserved. Is it possible after finite number of steps to switch positions of the pebbles?

Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).

Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$. (a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$. (b) Find a smaller $d$ (the smaller, the better) with the same property.