Olympiad problems

2008 Alexandru Myller, 1

$ 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]

2002 Spain Mathematical Olympiad, Problem 5

Tags: geometry , polygon , plane geometry , vector geometry

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 $\frac{2}{3}$.

2012 Gheorghe Vranceanu, 2

Tags: vector geometry , geometry , affine geometry

$ 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]

2010 Laurențiu Panaitopol, Tulcea, 4

Tags: vector geometry , geometry , parallelogram

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]

2017 Romania National Olympiad, 1

Tags: vector geometry , geometry , pure geometry , incenter , center of mass

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.

2017 District Olympiad, 2

Tags: vector geometry , geometry , circumcircle , incenter

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} . $

2019 Romania National Olympiad, 2

Tags: vector geometry , vector , geometry

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.

2019 Romania National Olympiad, 1

Tags: vector geometry , pure geometry , geometry , perpendicular bisector

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 $

2017 Romania National Olympiad, 2

Tags: vector geometry , pure geometry , geometry

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. $