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.

Let $ X: \equal{} \{x_1,x_2,\ldots,x_{29}\}$ be a set of $ 29$ boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules: [list]i) every boy play one and only one time against each other boy (so we can assume that every match has the form $ (x_i \text{ Vs } x_j)$ for some $ i \neq j$); ii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the win of the boy $ x_i$, then $ x_i$ gains $ 1$ point, and $ x_j$ doesn’t gain any point; iii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the parity of the two boys, then $ \frac {1}{2}$ point is assigned to both boys. [/list] (We assume for simplicity that in the imaginary match $ (x_i \text{ Vs } x_i)$ the boy $ x_i$ doesn’t gain any point). Show that for some positive integer $ k \le 29$ there exist a set of boys $ \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X$ such that, for all choice of the positive integer $ i \le 29$, the boy $ x_i$ gains always a integer number of points in the total of the matches $ \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}$. [i](Paolo Leonetti)[/i]

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

Consider a finite set of vectors in space $\{a_1, a_2, ... , a_n\}$ and the set $E$ of all vectors of the form $x=\sum_{i=1}^{n}{\lambda _i a_i}$, where $\lambda _i \in \mathbb{R}^{+}\cup \{0\}$. Let $F$ be the set consisting of all the vectors in $E$ and vectors parallel to a given plane $P$. Prove that there exists a set of vectors $\{b_1, b_2, ... , b_p\}$ such that $F$ is the set of all vectors $y$ of the form $y=\sum_{i=1}^{p}{\mu _i b_i}$, where $\mu _i \in \mathbb{R}^{+}\cup \{0\}$.

Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios. [b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram. [b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ? (Consecutive vertices of the parallelograms are labelled in alphabetical order.

Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.

In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.

Let $ABC$ be a triangle. Let $P, Q, R$ be the midpoints of $BC, CA, AB$ respectively. Let $U, V, W$ be the midpoints of $QR, RP, PQ$ respectively. Let $x=AU, y=BV, z=CW$. Prove that there exist a triangle with sides $x, y, z$.

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Let $f\colon \mathbb{R}^1\rightarrow \mathbb{R}^2$ be a continuous function such that $f(x)=f(x+1)$ for all $x$, and let $t\in [0,\frac14]$. Prove that there exists $x\in\mathbb{R}$ such that the vector from $f(x-t)$ to $f(x+t)$ is perpendicular to the vector from $f(x)$ to $f(x+\frac12)$. (translated by Miklós Maróti)

A river flows between two houses $ A$ and $ B$, the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from $ A$ to $ B$, using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.

Given a table with $2^n * n$ 1*1 squares ( $2^n$ rows and n column). In any square we put a number in {1, -1} such that no two rows are the same. Then we change numbers in some squares by 0. Prove that in new table we can choose some rows such that sum of all numbers in these rows equal to 0.

Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$. [i]J. Szucs[/i]

Let $ABC$ be a triangle. Suppose that the perpendicular bisector of $BC$ meets the circle of diameter $AB$ at a point $D$ at the opposite side of $BC$ with respect to $A$, and meets the circle through $A, C, D$ again at $E$. Prove that $\angle ACE=\angle BCD$. [i]Proposed by José Manuel Guerra and Victor Domínguez[/i]

Given two integers $n\geq 1$ and $q\geq 2$, let $A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}$. If $a=(a_1,\ldots ,a_n)$ and $b=(b_1,\ldots ,b_n)$ are two elements of $A$, let $\delta(a,b)=\#\{i:a_i\neq b_i\}$. Let further $t$ be a non-negative integer and $B$ a non-empty subset of $A$ such that $\delta(a,b)\geq 2t+1$, whenever $a$ and $b$ are distinct elements of $B$. Prove that the two statements below are equivalent: a) For any $a\in A$, there is a unique $b\in B$, such that $\delta (a,b)\leq t$; b) $\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n$

Show that for every odd integer $N>5$ there exist vectors $\bf u,v,w$ in (three-dimensional) space which are pairwise perpendicular, not parallel with any of the coordinate axes, have integer coordinates, and satisfy $N\bf =|u|=|v|=|w|.$ [i]Based on problem 2 of the 2018 Kürschák contest[/i]

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.

Let $n$ be a positive integer. For any positive integer $k$, let $0_k=diag\{\underbrace{0, ...,0}_{k}\}$ be a $k \times k$ zero matrix. Let $Y=\begin{pmatrix} 0_n & A \\ A^t & 0_{n+1} \end{pmatrix}$ be a $(2n+1) \times (2n+1)$ where $A=(x_{i, j})_{1\leq i \leq n, 1\leq j \leq n+1}$ is a $n \times (n+1)$ real matrix. Let $A^T$ be transpose matrix of $A$ i.e. $(n+1) \times n$ matrix, the element of $(j, i)$ is $x_{i, j}$. (a) Let complex number $\lambda$ be an eigenvalue of $k \times k$ matrix $X$. If there exists nonzero column vectors $v=(x_1, ..., x_k)^t$ such that $Xv=\lambda v$. Prove that 0 is the eigenvalue of $Y$ and the other eigenvalues of $Y$ can be expressed as a form of $\pm \sqrt{\lambda}$ where nonnegative real number $\lambda$ is the eigenvalue of $AA^t$. (b) Let $n=3$ and $a_1$, $a_2$, $a_3$, $a_4$ are $4$ distinct positive real numbers. Let $a=\sqrt[]{\sum_{1\leq i \leq 4}^{}a^{2}_{i}}$ and $x_{i,j}=a_i\delta_{i,j}+a_j\delta_{4,j}-\frac{1}{a^2}(a^2_{i}+a^2_{4})a_j$ where $1\leq i \leq 3, 1\leq j \leq 4$, $\delta_{i, j}= \begin{cases} 1 \text{ if } i=j\\ 0 \text{ if } i\neq j\\ \end{cases}\,$. Prove that $Y$ has 7 distinct eigenvalue.

Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.

The diagonals of the cyclic quadrilateral $ ABCD$ are intersecting at the point $ E$. $ K$ and $ M$ are the midpoints of $ AB$ and $ CD$, respectively. Let the points $ L$ on $ BC$ and $ N$ on $ AD$ s.t. $ EL\perp BC$ and $ EN\perp AD$.Prove that $ KM\perp LN$.

What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?

A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos(\angle AVB)$? $ \textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2} $

On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to $\text{(A)}\ 1.6 \qquad \text{(B)}\ 1.9 \qquad \text{(C)}\ 2.1 \qquad \text{(D)}\ 2.7 \qquad \text{(E)}\ 2.9$

Let $\Delta$ be a convex figure in a plane $\mathcal P$. Given a point $A\in\mathcal P$, to each pair $(M,N)$ of points in $\Delta$ we associate the point $m\in\mathcal P$ such that $\overrightarrow{Am}=\frac{\overrightarrow{MN}}2$ and denote by $\delta_A(\Delta)$ the set of all so obtained points $m$. (a) i. Prove that $\delta_A(\Delta)$ is centrally symmetric. ii. Under which conditions is $\delta_A(\Delta)=\Delta$? iii. Let $B,C$ be points in $\mathcal P$. Find a transformation which sends $\delta_B(\Delta)$ to $\delta_C(\Delta)$. (b) Determine $\delta_A(\Delta)$ if i. $\Delta$ is a set in the plane determined by two parallel lines. ii. $\Delta$ is bounded by a triangle. iii. $\Delta$ is a semi-disk. (c) Prove that in the cases $b.2$ and $b.3$ the lengths of the boundaries of $\Delta$ and $\delta_A(\Delta)$ are equal.

We consider three vectors drawn from the same initial point $O,$ of lengths $a,b$ and $c$, respectively. Let $E$ be the parallelepiped with vertex $O$ of which the given vectors are the edges and $H$ the parallelepiped with vertex $O$ of which the given vectors are the altitudes. Show that the product of the volumes of $E$ and $H$ equals $(abc)^{2}$ and generalize this result to $n$ dimensions.