On the sides $ AB,AC $ of a triangle $ ABC, $ consider the points $ M, $ respectively, $ N $ such that $ M\neq A\neq N $ and $ \frac{MB}{MA}\neq\frac{NC}{NA}. $ Show that the line $ MN $ passes through a point not dependent on $ M $ and $ N. $

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]

Danielle draws a point $O$ on the plane and a set of points $\mathcal P = \{P_0, P_1, \ldots , P_{2022}\}$ such that $$\angle{P_0OP_1} = \angle{P_1OP_2} = \cdots = \angle{P_{2021}OP_{2022}} = \alpha, \hspace{5pt} 0 < \alpha < \pi,$$where the angles are measured counterclockwise and for $0 \leq n \leq 2022$, $OP_n = r^n$, where $r > 1$ is a given real number. Then, obtain new sets of points in the plane by iterating the following process: given a set of points $\{A_0, A_1, \ldots , A_n\}$ in the plane, it is built a new set of points $\{B_0, B_1, \ldots , B_{n-1}\}$ such that $A_kA_{k+1}B_k$ is an equilateral triangle oriented clockwise for $0 \leq k \leq n - 1$. After carrying out the process $2022$ times from the set $P$, Danielle obtains a single point $X$. If the distance from $X$ to point $O$ is $d$, show that $$(r-1)^{2022} \leq d \leq (r+1)^{2022}.$$

Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.

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 $K$ be a convex planar set, symmetric about a point $O$, and let $X, Y , Z$ be three points in $K$. Show that $K$ contains the head of one of the vectors $\overrightarrow{OX} \pm \overrightarrow{OY} , \overrightarrow{OX} \pm \overrightarrow{OZ}, \overrightarrow{OY} \pm \overrightarrow{OZ}$.

Let be a point $ P $ in the interior of a triangle $ ABC $ such that $ BP=AC, M $ be the middlepoint of the segment $ AP, R $ be the middlepoint of $ BC $ and $ E $ be the intersection of $ BP $ with $ AC. $ Prove that the bisector of $ \angle BEA $ is perpendicular on $ MR $

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]

The cevians $ AP,BQ,CR $ of the triangle $ ABC $ are concurrent at $ F. $ Prove that the following affirmations are equivalent. $ \text{(i)} \overrightarrow{AP} +\overrightarrow{BQ} +\overrightarrow{CR} =0 $ $ \text{(ii)} F$ is the centroid of $ ABC $ [i]Doru Isac[/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 $ P $ be a point on the side $ AB $ of the triangle $ ABC. $ The parallels through $ P $ of the medians $ AA_1,BB_1 $ intersect $ BC,AC $ at $ R,Q, $ respectively. Show that $ P, $ the middlepoint of $ RQ $ and the centroid of $ ABC $ are collinear.

Let $ M,N,P $ on the sides $ AB,BC,CA, $ respectively, of a triangle $ ABC $ such that $ AM=BN=CP $ and such that $$ AB\cdot \overrightarrow{AT} +BC\cdot \overrightarrow{BT} +CA\cdot \overrightarrow{CT} =0, $$ where $ T $ is the centroid of $ MNP. $ Prove that $ ABC $ is equilateral.

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 $\angle A_iBA_j&gt;90^\circ$ holds for at least $n$ pairs $(i,j)$ with $\displaystyle{1\le i&lt;j\le n+1}$. Proposed by Géza Kós, Eötvös University, Budapest

$ O $ is the circumcentre of $ ABC $ and $ A_1\neq A $ is the point on $ AO $ and the circumcircle of $ ABC. $ The centers of mass of $ ABC, A_1BC $ are $ G,G_1, $ respectively, and $ P $ is the intersection of $ AG_1 $ with $ OG. $ Show that $ \frac{PG}{PO}=\frac{2}{3} . $ [i]Gabriel Popa, Paul Georgescu[/i]

Let $ABCDEF$ be a convex hexagon and denote $U$,$V$,$W$,$X$,$Y$ and $Z$ be the midpoint of $AB$,$BC$,$CD$,$DE$,$EF$ and $FA$ respectively. Prove that the length of $UX$,$VY$,$WZ$ can be the length of each sides of some triangle.

A spider built a web on the unit circle. The web is a planar graph with straight edges inside the circle, bounded by the circumference of the circle. Each vertex of the graph lying on the circle belongs to a unique edge, which goes perpendicularly inward to the circle. For each vertex of the graph inside the circle, the sum of the unit outgoing vectors along the edges of the graph is zero. Prove that the total length of the web is equal to the number of its vertices on the circle.

Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $ [i]Dragoș Lăzărescu[/i]

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.

On the sides $ AB $ and $ AC $ of the triangle $ ABC $ consider the points $ D, $ respectively, $ E, $ such that $$ \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . $$ If $ T $ is the intersection of $ DC $ and $ BE, $ determine the real number $ \alpha $ so that: $$ \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} . $$

[b]a)[/b] Prove that, for any point in the interior of a triangle, there are two points on the sides of this triangle such that the resultant of the vectors from the interior point those two points is the vector $ 0. $ [b]b)[/b] Prove that, for any point in the interior of a triangle, there are three points on the sides of this triangle such that the resultant of the vectors from the interior point those three points is the vector $ 0. $

Let $ ABC $ be a triangle in which $ O,I, $ are the circumcenter, respectively, incenter. The mediators of $ IA,IB,IC, $ form a triangle $ A_1B_1C_1. $ Show that $ \overrightarrow{OI}=\overrightarrow{OA_1} +\overrightarrow{OA_2} +\overrightarrow{OA_3} . $

Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality: $$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$ [I]Proposed by A. Khrabrov[/i]

Divide a $ 2\times 4 $ rectangle into $ 8 $ unit squares to obtain a set of $ 15 $ vertices denoted by $ \mathcal{M} . $ Find the points $ A\in\mathcal{M} $ that have the property that the set $ \mathcal{M}\setminus \{ A\} $ can form $ 7 $ pairs $ \left( A_1,B_1\right) ,\left( A_2,B_2\right) ,\ldots ,\left( A_7,B_7\right)\in\mathcal{M}\times\mathcal{M} $ such that $$ \overrightarrow{A_1B_1} +\overrightarrow{A_2B_2} +\cdots +\overrightarrow{A_7B_7} =\overrightarrow{O} . $$

[b]a)[/b] Let $ AA',BB',CC' $ be the altitudes of a triangle $ ABC. $ Prove that $$ \frac{BC}{AA'}\cdot \overrightarrow{AA'} +\frac{AC}{BB'}\cdot \overrightarrow{BB'} +\frac{AB}{CC'}\cdot \overrightarrow{CC'} =0. $$ [b]b)[/b] The sum of the vectors that are perpendicular to the sides of a convex polygon and have equal lengths as those sides, respectively, is $ 0. $

On the sides (excluding its endpoints) $ AB,BC,CD,DA $ of a parallelogram consider the points $ M,N,P,Q, $ respectively, such that $ \overrightarrow{AP} +\overrightarrow{AN} +\overrightarrow{CQ} +\overrightarrow{CM} = 0. $ Show that $ QN, PM,AC $ are concurrent. [i]Adrian Ivan[/i]