Find all natural numbers which are the cardinal of a set of nonzero Euclidean vectors whose sum is $ 0, $ the sum of any two of them is nonzero, and their magnitudes are equal.

Given some points in the plane. For some pairs $A,B$ the vector $AB$ is chosen. For every point the number of the chosen vectors starting in that point equal to the number of the chosen vectors ending in that point. Prove that the sum of the chosen vectors equals to zero vector.

Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

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)

Do there exist $4$ vectors in the plane so that none is a multiple of another, but the sum of each pair is perpendicular to the sum of the other two? Do there exist $91$ non-zero vectors in the plane such that the sum of any $19$ is perpendicular to the sum of the others?

Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.

In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two $2011\times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk}).$ Initially, $T=W=0.$ After every game, for every $(h,k)$ (including for $h=k),$ if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1,$ while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1.$ Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative integer divisible by $2^{2010}.$

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).

Let $n\ge2$, let $A_1,A_2,\ldots,A_{n+1}$ be $n+1$ points in the $n$-dimensional Euclidean space, not lying on the same hyperplane, and let $B$ be a point strictly inside the convex hull of $A_1,A_2,\ldots,A_{n+1}$. Prove that $\angle A_iBA_j>90^\circ$ holds for at least $n$ pairs $(i,j)$ with $\displaystyle{1\le i<j\le n+1}$. Proposed by Géza Kós, Eötvös University, Budapest

A [i]tanned vector[/i] is a nonzero vector in $\mathbb R^3$ with integer entries. Prove that any tanned vector of length at most $2021$ is perpendicular to a tanned vector of length at most $100$. [i]Proposed by Ethan Tan[/i]

Let be a square $ ABCD, $ a point $ E $ on $ AB, $ a point $ N $ on $ CD, $ points $ F,M $ on $ BC, $ name $ P $ the intersection of $ AN $ with $ DE, $ and name $ Q $ the intersection of $ AM $ with $ EF. $ If the triangles $ AMN $ and $ DEF $ are equilateral, prove that $ PQ=FM. $

Let $U$ denote the set $\{ f\in C[0, 1] \colon |f(x)|\leq 1\, \mathrm{for}\,\mathrm{all}\, x\in [0, 1]\}$. Prove that there is no topology on $C[0, 1]$ that, together with the linear structure of $C[0,1]$, makes $C[0,1]$ into a topological vector space in which the set $U$ is compact. (Assume that topological vector spaces are Hausdorff) [V. Totik]

Triangles $ ABC$ and $ ADE$ have areas $ 2007$ and $ 7002,$ respectively, with $ B \equal{} (0,0),$ $ C \equal{} (223,0),$ $ D \equal{} (680,380),$ and $ E \equal{} (689,389).$ What is the sum of all possible x-coordinates of $ A?$ $ \textbf{(A)}\ 282 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 900 \qquad \textbf{(E)}\ 1200$

On the sides $BC$, $CA$, $AB$ of an acute-angled triangle $ABC$, we erect (outwardly) the squares $BB_aC_aC$, $CC_bA_bA$, $AA_cB_cB$, respectively. On the sides $B_cB_a$ and $C_aC_b$ of the triangles $BB_cB_a$ and $CC_aC_b$, we erect (outwardly) the squares $B_cB_vB_uB_a$ and $C_aC_uC_vC_b$. Prove that $B_uC_u\parallel BC$. [i]Comment.[/i] This problem originates in the 68th Moscow MO 2005, and a solution was posted in http://www.mathlinks.ro/Forum/viewtopic.php?t=30184 . However ingenious this solution is, there is a different one which shows a bit more: $B_uC_u=4\cdot BC$. Darij

Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$, the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular. [i]Alternative formulation.[/i] The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$ and $AB$ in $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$, the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$, and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.