Olympiad problems

2012 District Olympiad, 3

Let $ABC$ be an acute triangle. Consider the points $M, N \in (BC), Q \in (AB), P \in (AC)$ such that the $MNPQ$ is a rectangle. Prove that if the center of the rectangle $MNPQ$ coincides with the center of gravity of the triangle $ABC$, then $AB = AC = 3AP$

2006 Mathematics for Its Sake, 2

Tags: geometry , cevian , centroid , center of mass , vector geometry

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]

2015 Sharygin Geometry Olympiad, 7

Tags: angle , orthocenter , centroid , geometry

Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB' $meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$. (D. Krekov)

2024 Yasinsky Geometry Olympiad, 4

Tags: geometry , construction , incenter , centroid , parallel

Let $ I $ and $ M $ be the incenter and the centroid of a scalene triangle $ ABC $, respectively. A line passing through point $ I $ parallel to $ BC $ intersects $ AC $ and $ AB $ at points $ E $ and $ F $, respectively. Reconstruct triangle $ ABC $ given only the marked points $ E, F, I, $ and $ M $. [i]Proposed by Hryhorii Filippovskyi[/i]

2003 Oral Moscow Geometry Olympiad, 4

Tags: incenter , centroid , perpendicular , geometry

In triangle $ABC$, $M$ is the point of intersection of the medians, $O$ is the center of the inscribed circle, $A', B', C'$ are the touchpoints with the sides $BC, CA, AB$, respectively. Prove that if $CA'= AB$, then $OM$ and $AB$ are perpendicular. PS. There is a a typo

2025 Romania National Olympiad, 1

Tags: geometry , reflection , circumcircle , centroid , complex number geometry

Let $M$ be a point in the plane, distinct from the vertices of $\triangle ABC$. Consider $N,P,Q$ the reflections of $M$ with respect to lines $AB, BC$ and $CA$, in this order. a) Prove that $N, P ,Q$ are collinear if and only if $M$ lies on the circumcircle of $\triangle ABC$. b) If $M$ does not lie on the circumcircle of $\triangle ABC$ and the centroids of triangles $\triangle ABC$ and $\triangle NPQ$ coincide, prove that $\triangle ABC$ is equilateral.

1958 Polish MO Finals, 2

Tags: geometry , centroid , center of gravity

Each side of a convex quadrilateral $ ABCD $ is divided into three equal parts; a straight line is drawn through the dividing points of sides $ AB $ and $ AD $ that lie closer to vertex $ A $, and similarly for vertices $ B $, $ C $, $ D $. Prove that the center of gravity of the quadrilateral formed by the drawn lines coincides with the center of gravity of quadrilateral $ ABCD $.

2005 Sharygin Geometry Olympiad, 5

Tags: geometry , locus , orthocenter , circumcenter , centroid , circumcircle

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.

1975 All Soviet Union Mathematical Olympiad, 213

Tags: geometry , perimeter , centroid

Three flies are crawling along the perimeter of the triangle $ABC$ in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the triangle $ABC$ . (The centre of masses for the triangle is the point of medians intersection.

2018 Ramnicean Hope, 2

Tags: centroid , center of mass , geometry

Let be the points $ M,N,P, $ on the sides $ BC,AC,AB $ (not on their endpoints), respectively, of a triangle $ ABC, $ such that $ \frac{BM}{MC} =\frac{CN}{NA} =\frac{AP}{PB} . $ Denote $ G_1,G_2,G_3 $ the centroids of $ APN,BMP,CNM, $ respectively. Show that the $ MNP $ has the same centroid as $ G_1G_2G_3. $ [i]Ovidiu Țâțan[/i]

1997 All-Russian Olympiad Regional Round, 11.2

Tags: reflection , centroid , geometry

All vertices of triangle $ABC$ lie inside square $K$. Prove that if all of them are reflected symmetrically with respect to the point of intersection of the medians of triangle $ABC$, then at least one of the resulting three points will be inside $K$.

2016 Irish Math Olympiad, 6

Tags: centroid , right angle , geometry

Triangle $ABC$ has sides $a = |BC| > b = |AC|$. The points $K$ and $H$ on the segment $BC$ satisfy $|CH| = (a + b)/3$ and $|CK| = (a - b)/3$. If $G$ is the centroid of triangle $ABC$, prove that $\angle KGH = 90^o$.

2019 Bulgaria EGMO TST, 1

Tags: euler , euler line , centroid , circumcenter , geometry , circumcircle

Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.)

Kyiv City MO 1984-93 - geometry, 1992.8.3

Tags: geometry , centroid , locus

Find the locus of the intersection points of the medians all triangles inscribed in a given circle.

2021 Dutch IMO TST, 2

Tags: right angle , centroid , right triangle , geometry

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

1978 Germany Team Selection Test, 1

Tags: analytic geometry , combinatorics , point set , centroid

Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$