Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

A set $M$ consists of $n$ elements. Find the greatest $k$ for which there is a collection of $k$ subsets of $M$ such that for any subsets $A_{1},...,A_{j}$ from the collection, there is an element belonging to an odd number of them

Let’s consider three pairwise non-parallel straight constant lines in the plane. Three points are moving along these lines with different nonzero velocities, one on each line (we consider the movement to have taken place for infinite time and continue infinitely in the future). Is it possible to determine these straight lines, the velocities of each moving point and their positions at some “zero” moment in such a way that the points never were, are or will be collinear?

We say that a set $\displaystyle A$ of non-zero vectors from the plane has the property $\displaystyle \left( \mathcal S \right)$ iff it has at least three elements and for all $\displaystyle \overrightarrow u \in A$ there are $\displaystyle \overrightarrow v, \overrightarrow w \in A$ such that $\displaystyle \overrightarrow v \neq \overrightarrow w$ and $\displaystyle \overrightarrow u = \overrightarrow v + \overrightarrow w$. (a) Prove that for all $\displaystyle n \geq 6$ there is a set of $\displaystyle n$ non-zero vectors, which has the property $\displaystyle \left( \mathcal S \right)$. (b) Prove that every finite set of non-zero vectors, which has the property $\displaystyle \left( \mathcal S \right)$, has at least $\displaystyle 6$ elements. [i]Mihai Baluna[/i]

Let $n$ be a positive integer. Suppose we are given $2^n+1$ distinct sets, each containing finitely many objects. Place each set into one of two categories, the red sets and the blue sets, so that there is at least one set in each category. We define the [i]symmetric difference[/i] of two sets as the set of objects belonging to exactly one of the two sets. Prove that there are at least $2^n$ different sets which can be obtained as the symmetric difference of a red set and a blue set.

Let $k$ and $N$ be positive real numbers which satisfy $k\leq N$. For $1\leq i \leq k$, there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property. For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \} $, $A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}$ is not an empty set. Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \} $ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k$. ($A \triangle B=A \cup B-A \cap B$)

Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.

There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week.

We define a [i]pseudo-inverse[/i] $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations \[ A = ABA \quad \text{ and } \quad B=BAB. \] a) Prove that any square matrix has at least a pseudo-inverse. b) For which matrix $A$ is the pseudo-inverse unique? [i]Marius Cavachi[/i]

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.

Let $v_1, v_2, \ldots, v_m$ be vectors in $\mathbb{R}^n$, such that each has a strictly positive first coordinate. Consider the following process. Start with the zero vector $w = (0, 0, \ldots, 0) \in \mathbb{R}^n$. Every round, choose an $i$ such that $1 \le i \le m$ and $w \cdot v_i \le 0$, and then replace $w$ with $w + v_i$. Show that there exists a constant $C$ such that regardless of your choice of $i$ at each step, the process is guaranteed to terminate in (at most) $C$ rounds. The constant $C$ may depend on the vectors $v_1, \ldots, v_m$.

Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.

Find the sum of the indexes of the singular points other than zero of the vector field \[z\overline{z}^2+z^4+2\overline{z}^4\]

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.

There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps. what does barycenter of n distinct points mean?

Prove that these two statements are equivalent for an $n$ dimensional vector space $V$: [b]$\cdot$[/b] For the linear transformation $T:V\longrightarrow V$ there exists a base for $V$ such that the representation of $T$ in that base is an upper triangular matrix. [b]$\cdot$[/b] There exist subspaces $\{0\}\subsetneq V_1 \subsetneq ...\subsetneq V_{n-1}\subsetneq V$ such that for all $i$, $T(V_i)\subseteq V_i$.

a) Let $x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R}$ and $a_{ij}=\sin(x_i-y_j),\ i,j=\overline{1,3}$ and $A=(a_{ij})\in \mathcal{M}_3$ Prove that $\det A=0$. b) Let $z_1,z_2,\ldots,z_{2n}\in \mathbb{C}^*,\ n\ge 3$ such that $|z_1|=|z_2|=\ldots=|z_{n+3}|$ and $\arg z_1\ge \arg z_2\ge \ldots\ge \arg(z_{n+3})$. If $b_{ij}=|z_i-z_{j+n}|,\ i,j=\overline{1,n}$ and $B=(b_{ij})\in \mathcal{M}_n$, prove that $\det B=0$.

Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2=\sqrt{11}$. Find $\sum^4_{k=1}(CE_k)^2$.

A particle moves on a circle with center $O$, starting from rest at a point $P$ and coming to rest again at a point $Q$, without coming to rest at any intermediate point. Prove that the acceleration vector of the particle does not vanish at any point between $P$ and $ Q$ and that, at some point $R$ between $P$ and $Q$, the acceleration vector points in along the radius $RO.$

Let $ABC$ be a triangle. Let $A_1$ and $A_2$ be points on side $BC, B_1$ and $B_2$ be points on side $CA$ and $C_1$ and $C_2$ be points on side $AB$ such that $A_1A_2B_1B_2C_1C_2$ is a convex hexagon and that $B,A_1,A_2$ and $C$ are located in that order on side $BC$. We say that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if there exists a triangle $PQR$ and there exist $X,Y$ and $Z$ on sides $QR, RP$ and $PQ$ respectively, such that triangle $AB_2C_1$ is congruent in that order to triangle $PYZ$, triangle $BA_1C_2$ is congruent in that order to triangle $QXZ$ and triangle $CA_2B_1$ is congruent in that order to triangle $RXY$. Prove that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if and only if the centroids of triangles $A_1B_1C_1$ and $A_2B_2C_2$ coincide.

$AB$ is a chord of length $6$ of a circle centred at $O$ and of radius $5$. Let $PQRS$ denote the square inscribed in the sector $OAB$ such that $P$ is on the radius $OA$, $S$ is on the radius $OB$ and $Q$ and $R$ are points on the arc of the circle between $A$ and $B$. Find the area of $PQRS$.

XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external tangent to these circles is tangent to the constant circle( ditermine the radius and the locus of its center).

Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB=12;$ face $ABFG$ is a trapezoid with $\overline{AB}$ parallel to $\overline{GF},$ $BF=AG=8,$ and $GF=6;$ and face $CDE$ has $CE=DE=14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is 12. Given that $EG^2=p-q\sqrt{r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$

Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \] [i]selected by Mircea Lascu[/i]