Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$

Let $d>1$ be integer and $0<r<\frac12$. Show that there exist finitely many (depending only on $d,r$) nonzero vectors in $\mathbb{R}^d$ such that if the distance of a straight line in $\mathbb{R}^d$ from the integer lattice $\mathbb{Z}^d$ is at least $r$, then this line is orthogonal to one of these finitely many vectors. (translated by L. Erdős)

Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$ $$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$ Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}$.

In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AC$ and $BC$ respectively. The distance from the midpoint of $BD$ to the midpoint of $AE$ is $4.5$. What is the length of side $AB$?

Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

Let $ x_1,x_2,..,x_n$ be non-negative numbers whose sum is $ 1$ . Show that there exist numbers $ a_1,a_2,\ldots ,a_n$ chosen from amongst $ 0,1,2,3,4$ such that $ a_1,a_2,\ldots ,a_n$ are different from $ 2,2,\ldots ,2$ and $ 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}$.

Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S\subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$. Say that a vector subspace $T\subseteq \mathbb{M}$ is a $\emph{covering matrix space}$ if \[ \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p \] Such a $T$ is minimal if it doesn't contain a proper vector subspace $S\subset T$ such that $S$ is also a covering matrix space. [list] (a) (8 points) Let $T$ be a minimal covering matrix space and let $n=\dim (T)$ Prove that \[ \delta(T)\le \dbinom{n}{2} \] (b) (2 points) Prove that for every integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T=n$ and $\delta(T)=\dbinom{n}{2}$[/list]

A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation $x=f(t)$ (where $f(t)$ is not constant) gives the horizontal distance in terms of the time $t$. Show that the vertical distance $y$ is given by $$y=-gf(t) \int \frac{dt}{f'(t)} + g \int \frac{f(t)}{f'(t)} \, dt +Af(t)+B$$ where $A$ and $B$ are constants and $g$ is the acceleration due to gravity.

Let $ A$ and $ B$ be nonsingular matrices of order $ p$, and let $ \xi$ and $ \eta$ be independent random vectors of dimension $ p$. Show that if $ \xi,\eta$ and $ \xi A\plus{} \eta B$ have the same distribution, if their first and second moments exist, and if their covariance matrix is the identity matrix, then these random vectors are normally distributed. [i]B. Gyires[/i]

Introduce a standard scalar product in $\mathbb{R}^4.$ Let $V$ be a partial vector space in $\mathbb{R}^4$ produced by $\left( \begin{array}{c} 1 \\ -1 \\ -1 \\ 1 \end{array} \right),\left( \begin{array}{c} 1 \\-1 \\ 1 \\ -1 \end{array} \right).$ Find a pair of base of orthogonal complement $W$ for $V$ in $\mathbb{R}^4.$

In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$. [i]Author: Ray Li[/i]

Let $a$ and $b$ be real numbers satisfying $2(\sin a + \cos a) \sin b = 3 - \cos b$. Find $3 \tan^2a+4\tan^2 b$.

Find the smallest positive real number $r$ with the following property: For every choice of $2023$ unit vectors $v_1,v_2, \dots ,v_{2023} \in \mathbb{R}^2$, a point $p$ can be found in the plane such that for each subset $S$ of $\{1,2, \dots , 2023\}$, the sum $$\sum_{i \in S} v_i$$ lies inside the disc $\{x \in \mathbb{R}^2 : ||x-p|| \leq r\}$.

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be unit vectors in $R^3$. Prove that $$\sqrt{1-\overrightarrow{a}\cdot\overrightarrow{b}}\le \sqrt{1-\overrightarrow{a}\cdot\overrightarrow{c}}+\sqrt{1-\overrightarrow{c}\cdot\overrightarrow{b}}$$ (A.Mirotin)

If $a, b, c$ are non-negative real numbers with $ a^2 \plus{} b^2 \plus{} c^2 \equal{} 1$, prove that: \[ \frac {a}{b^2 \plus{} 1} \plus{} \frac {b}{c^2 \plus{} 1} \plus{} \frac {c}{a^2 \plus{} 1} \geq \frac {3}{4}(a\sqrt {a} \plus{} b\sqrt {b} \plus{} c\sqrt {c})^2\]

Prove that there is no finite set which contains more than $ 2N,$ with $ N > 3,$ pairwise non-collinear vectors of the plane, and to which the following two characteristics apply: 1) for $ N$ arbitrary vectors from this set there are always further $ N\minus{}1$ vectors from this set so that the sum of these is $ 2N\minus{}1$ vectors is equal to the zero-vector; 2) for $ N$ arbitrary vectors from this set there are always further $ N$ vectors from this set so that the sum of these is $ 2N$ vectors is equal to the zero-vector.

In the triangle $ABC$ let $D,E$ and $F$ be the mid-points of the three sides, $X,Y$ and $Z$ the feet of the three altitudes, $H$ the orthocenter, and $P,Q$ and $R$ the mid-points of the line segment joining $H$ to the three vertices. Show that the nine points $D,E,F,P,Q,R,X,Y,Z$ lie on a circle.

Given two equilateral triangles $A_1B_1C_1$ and $A_2B_2C_2$ in the plane. (The vertices are mentioned counterclockwise.) We draw vectors $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$, from the arbitrary point $O$, equal to $\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}$ respectively. Prove that the triangle $ABC$ is equilateral.

Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !

Let $m,n$ be positive integers with $m \ge n$. Let $S$ be the set of pairs $(a,b)$ of relatively prime positive integers such that $a,b \le m$ and $a+b > m$. For each pair $(a,b)\in S$, consider the nonnegative integer solution $(u,v)$ to the equation $au - bv = n$ chosen with $v \ge 0$ minimal, and let $I(a,b)$ denote the (open) interval $(v/a, u/b)$. Prove that $I(a,b) \subseteq (0,1)$ for every $(a,b)\in S$, and that any fixed irrational number $\alpha\in(0,1)$ lies in $I(a,b)$ for exactly $n$ distinct pairs $(a,b)\in S$. [i]Victor Wang, inspired by 2013 ISL N7[/i]

11.6 Construct for each vertex of the quadrilateral of area $S$ a symmetric point wrt to the diagonal, which doesn't contain this vertex. Let $S'$ be an area of the obtained quadrilateral. Prove that $\frac{S'}{S}<3$. ([i]L. Emel'yanov[/i])

$20$ points with no three collinear are given. How many obtuse triangles can be formed by these points? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 2{{10}\choose{3}} \qquad \textbf{(D)}\ 3{{10}\choose{3}} \qquad \textbf{(E)}\ {{20}\choose{3}}$

$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.

Given a real number $p>1$, a continuous function $h\colon [0,\infty)\to [0,\infty)$, and a smooth vector field $Y\colon \mathbb{R}^n \to \mathbb{R}^n$ with $\mathrm{div}~Y=0$, prove the following inequality \[\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.\]