Consider a triangle $ ABC $ having incenter $ I $ and inradius $ r. $ Let $ D $ be the tangency of $ ABC $ 's incircle with $ BC, $ and $ E $ on the line $ BC $ such that $ AE $ is perpendicular to $ BC, $ and $ M\neq E $ on the segment $ AE $ such that $ AM=r. $ [b]a)[/b] Give an idenity for $ \frac{BD}{DC} $ involving only the lengths of the sides of the triangle. [b]b)[/b] Prove that $ AB \cdot \overrightarrow{IC} +BC\cdot \overrightarrow{IA} +CA\cdot \overrightarrow{IB} =0. $ [b]c)[/b] Show that $ MI $ passes through the middle of the side $ BC. $ [i]Cătălin Zârnă[/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]

Let $ C_1,C_2 $ be two distinct concentric circles, and $ BA $ be a diameter of $ C_1. $ Choose the points $ M,N $ on $ C_1,C_2, $ respectively, but not on the line $ BA. $ [b]a)[/b] Show that there are unique points $ P,Q $ on $ MA,MB, $ respectively, so that $ N $ is the middle of $ PQ. $ [b]b)[/b] Prove that the value $ AP^2+BQ^2 $ does not depend on $ M,N. $ [i]Mihai Piticari, Mihail Bălună[/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.

Let $\triangle ABC$ be an acute-angled triangle, with circumcenter $O$, circumradius $R$ and orthocenter $H$. Let $A_1$ be a point on $BC$ such that $A_1H+A_1O=R$. Define $B_1$ and $C_1$ similarly. If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$, prove that $\triangle ABC$ is equilateral.

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]

Let $ABCD$ be a parallelogram of center $O$. Prove that for any point $M\in (AB)$, there exist unique points $N\in (OC)$ and $P\in (OD)$ such that $O$ is the center of mass of $\triangle MNP$.

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]