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.

Let $d\leq n$ be positive integers and $A$ a real $d\times n$ matrix. Let $\sigma(A)$ be the supremum of $\inf_{v\in W,|v|=1}|Av|$ over all subspaces $W$ of $R^n$ with dimension $d$. For each $j \leq d$, let $r(j) \in \mathbb{R}^n$ be the $j$th row vector of $A$. Show that: \[\sigma(A) \leq \min_{i\leq d} d(r(i), \langle r(j), j\ne i\rangle) \leq \sqrt{n}\sigma(A)\] In which all are euclidian norms and $d(r(i), \langle r(j), j\ne i\rangle)$ denotes the distance between $r(i)$ and the span of $r(j), 1 \leq j \leq d, j\ne i$.

Let $ Q $ be a point, $ H,O $ be the orthocenter and circumcenter, respectively, of a triangle $ ABC, $ and $ D,E,F, $ be the symmetric points of $ Q $ with respect to $ A,B,C, $ respectively. Also, $ M,N,P $ are the middle of the segments $ AE,BF,CD, $ and $ G,G',G'' $ are the centroids of $ ABC,MNP,DEF, $ respectively. Prove the following propositions: [b]a)[/b] $ \frac{1}{2}\overrightarrow{OG} =\frac{1}{3}\overrightarrow{OG'}=\frac{1}{4}\overrightarrow{OG''} $ [b]b)[/b] $ Q=O\implies \overrightarrow{OG'} =\overrightarrow{G'H} $ [b]c)[/b] $ Q=H\implies G'=O $ [i]Cătălin Zîrnă[/i]

Consider the triangle $ABC{}$ and let $I_A{}$ be its $A{}$-excenter. Let $M,N$ and $P{}$ be the projections of $I_A{}$ onto the lines $AC,BC{}$ and $AB{}$ respectively. Prove that if $\overrightarrow{I_AM}+\overrightarrow{I_AP}=\overrightarrow{I_AN}$ then $ABC{}$ is an equilateral triangle.

$ A',B',C' $ are the feet of the heights of an acute-angled triangle $ ABC. $ Calculate $$ \frac{\text{area} (ABC)}{\text{area}\left( A'B'C'\right)} , $$ knowing that $ ABC $ and $ A'B'C' $ have the same center of mass. [i]Carmen[/i] and [i]Viorel Botea[/i]

Let $ H $ denote the orthocenter of an acute triangle $ ABC, $ and $ A_1,A_2,A_3 $ denote the intersections of the altitudes of this triangle with its circumcircle, and $ A',B',C' $ denote the projections of the vertices of this triangle on their opposite sides. [b]a)[/b] Prove that the sides of the triangle $ A'B'C' $ are parallel to the sides of $ A_1B_1C_1. $ [b]b)[/b] Show that $ B_1C_1\cdot\overrightarrow{HA_1} +C_1A_1\cdot\overrightarrow{HB_1} +A_1B_1\cdot\overrightarrow{HC_1} =0. $ [i]Geoghe Duță[/i]

$ G $ is the centroid of $ ABC. $ The incircle of $ ABC $ touches $ BC,CA,AB $ at $ D,E,F, $ respectively. Show that $ ABC $ is equilateral if and only if $ BC\cdot\overrightarrow{GD}+ AC\cdot\overrightarrow{GE} +AB\cdot\overrightarrow{GF} =0. $ [i]Marian Ursărescu[/i]

In a quadrilateral $ABCD$ let $E$ be the intersection of the two diagonals, I the center of the parallelogram whose vertices are the midpoints of the four sides of the quadrilateral, and K the center of the parallelogram whose sides pass through the points. divide the four sides of the quadrilateral into three equal parts (see illustration ). [img]https://cdn.artofproblemsolving.com/attachments/1/c/8f2617103edd8361b8deebbee13c6180fa848b.png[/img] a) Prove that $\overrightarrow{EK} =\frac43 \overrightarrow{EI}$. b) Prove that $$\lambda_A \overrightarrow{KA} +\lambda_B \overrightarrow{KB} + \lambda_C \overrightarrow{KC} + \lambda_D \overrightarrow{KD} = \overrightarrow{0}$$ , where $$\lambda_A=1+\frac{S(ADB)}{S(ABCD)},\lambda_B=1+\frac{S(BCA)}{S(ABCD)},\lambda_C=1+\frac{S(CDB)}{S(ABCD)},\lambda_D=1+\frac{S(DAC)}{S(ABCD)}$$ , where $S$ is the area symbol.

Consider $2002$ segments on a plane, such that their lengths are the same. Prove that there exists such a straight line $r$ such that the sum of the lengths of the projections of the $2002$ segments about $r$ is less than $\frac{2}{3}$.