Olympiad problems

1979 Swedish Mathematical Competition, 6

Find the sharpest inequalities of the form $a\cdot AB < AG < b\cdot AB$ and $c\cdot AB < BG < d\cdot AB$ for all triangles $ABC$ with centroid $G$ such that $GA > GB > GC$.

1949 Moscow Mathematical Olympiad, 167

Tags: geometry , centroid , hexagon

The midpoints of alternative sides of a convex hexagon are connected by line segments. Prove that the intersection points of the medians of the two triangles obtained coincide.

Estonia Open Junior - geometry, 2018.2.5

Tags: centroid , geometric inequality , area , geometry

Medians $AD, BE$, and $CF$ of triangle $ABC$ intersect at point $M$. Is it possible that the circles with radii $MD, ME$, and $MF$ a) all have areas smaller than the area of triangle $ABC$, b) all have areas greater than the area of triangle $ABC$, c) all have areas equal to the area of triangle $ABC$?

2007 Sharygin Geometry Olympiad, 5

Tags: centroid , acute , median , geometry

Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$. Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.

2016 Irish Math Olympiad, 6

Tags: geometry , centroid , right angle

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 $\angle KGH = 90^o$.

1990 Mexico National Olympiad, 5

Tags: lattice points , centroid , combinatorial geometry , geometry

Given $19$ points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates.

2005 Switzerland - Final Round, 1

Tags: geometry , equilateral , centroid , concyclic , median , cyclic quadrilateral

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

1936 Eotvos Mathematical Competition, 2

Tags: equal area , centroid , geometry

$S$ is a point inside triangle $ABC$ such that the areas of the triangles $ABS$, $BCS$ and $CAS$ are all equal. Prove that $S$ is the centroid of $ABC$.

2015 Saudi Arabia GMO TST, 3

Tags: symmetry , geometry , centroid , concurrency

Let $ABC$ be a triangle and $G$ its centroid. Let $G_a, G_b$ and $G_c$ be the orthogonal projections of $G$ on sides $BC, CA$, respectively $AB$. If $S_a, S_b$ and $S_c$ are the symmetrical points of $G_a, G_b$, respectively $G_c$ with respect to $G$, prove that $AS_a, BS_b$ and $CS_c$ are concurrent. Liana Topan

2017 Bundeswettbewerb Mathematik, 3

Tags: centroid , perpendicular , incenter , geometry

Given is a triangle with side lengths $a,b$ and $c$, incenter $I$ and centroid $S$. Prove: If $a+b=3c$, then $S \neq I$ and line $SI$ is perpendicular to one of the sides of the triangle.

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]

2023 Baltic Way, 14

Tags: geometry , orthogonalaty , centroid

Let $ABC$ be a triangle with centroid $G$. Let $D, E, F$ be the circumcenters of triangles $BCG, CAG, ABG$. Let $X$ be the intersection of the perpendiculars from $E$ to $AB$ and from $F$ to $AC$. Prove that $DX$ bisects $EF$.

2010 Saudi Arabia Pre-TST, 1.4

Tags: geometry , centroid , collinear

In triangle $ABC$ with centroid $G$, let $M \in (AB)$ and $N \in (AC)$ be points on two of its sides. Prove that points $M, G, N$ are collinear if and only if $\frac{MB}{MA}+\frac{NC}{NA}=1$.

2014 Chile National Olympiad, 2

Tags: parallelogram , area , geometry , centroid

Consider an $ABCD$ parallelogram of area $1$. Let $E$ be the center of gravity of the triangle $ABC, F$ the center of gravity of the triangle $BCD, G$ the center of gravity of the triangle $CDA$ and $H$ the center of gravity of the triangle $DAB$. Calculate the area of quadrilateral $EFGH$.

Estonia Open Junior - geometry, 2008.1.3

Tags: similar triangles , geometry , centroid

Let $M$ be the intersection of the medians $ABC$ of the triangle and the midpoint of the side $BC$. $A$ line parallel to side $BC$ and passing through point $M$ intersects sides $AB$ and $AC$ at points $X$ and $Y$ respectively. Let the point of intersection of the lines $XC$ and $MB$ be $Q$ and let $P$ intersection point of the lines $YB$ and $MC$ be $P$ . Prove that the triangles $DPQ$ and $ABC$ are similar.

2020 Ukrainian Geometry Olympiad - December, 5

Tags: circumcircle , centroid , circumcenter , parallel , geometry

Let $ABC$ be an acute triangle with $\angle ACB = 45^o$, $G$ is the point of intersection of the medians, and $O$ is the center of the circumscribed circle. If $OG =1$ and $OG \parallel BC$, find the length of $BC$.

2011 Sharygin Geometry Olympiad, 13

Tags: tetrahedron , locus , centroid , inscribed triangle , 3d geometry , geometry

a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex). b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).

1978 Germany Team Selection Test, 1

Tags: analytic geometry , point set , centroid , combinatorics

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 $n.$

2012 Belarus Team Selection Test, 1

Tags: ratio , orthocenter , centroid , incenter , geometry

For any point $X$ inside an acute-angled triangle $ABC$ we define $$f(X)=\frac{AX}{A_1X}\cdot \frac{BX}{B_1X}\cdot \frac{CX}{C_1X}$$ where $A_1, B_1$, and $C_1$ are the intersection points of the lines $AX, BX,$ and $CX$ with the sides $BC, AC$, and $AB$, respectively. Let $H, I$, and $G$ be the orthocenter, the incenter, and the centroid of the triangle $ABC$, respectively. Prove that $f(H) \ge f(I) \ge f(G)$ . (D. Bazylev)

2024 Romania National Olympiad, 2

Tags: geometry , circumcircle , inscriptible , centroid , pentagon

We consider the inscriptible pentagon $ABCDE$ in which $AB=BC=CD$ and the centroid of the pentagon coincides with the circumcenter. Prove that the pentagon $ABCDE$ is regular. [i]The centroid of a pentagon is the point in the plane of the pentagon whose position vector is equal to the average of the position vectors of the vertices.[/i]

2010 Saudi Arabia IMO TST, 1

Tags: perpendicular , centroid , geometry

In triangle $ABC$ the circumcircle has radius $R$ and center $O$ and the incircle has radius $r$ and center $I\ne O$ . Let $G$ denote the centroid of triangle $ABC$. Prove that $IG \perp BC$ if and only if $AB = AC$ or $AB + AC = 3BC$.

Kyiv City MO Seniors 2003+ geometry, 2004.10.5

Tags: orthocenter , geometry , centroid

Let the points $M$ and $N$ in the triangle $ABC$ be the midpoints of the sides $BC$ and $AC$, respectively. It is known that the point of intersection of the altitudes of the triangle $ABC$ coincides with the point of intersection of the medians of the triangle $AMN$. Find the value of the angle $ABC$.

2013 Saudi Arabia Pre-TST, 1.4

Tags: cyclic quadrilateral , centroid , geometry

$ABC$ is a triangle, $G$ its centroid and $A',B',C'$ the midpoints of its sides $BC,CA,AB$, respectively. Prove that if the quadrilateral $AC'GB'$ is cyclic then $AB \cdot CC' = AC \cdot BB'$:

1974 Chisinau City MO, 71

Tags: geometry , centroid , triangle construction

The sides of the triangle $ABC$ lie on the sides of the angle $MAN$. Construct a triangle $ABC$ if the point $O$ of the intersection of its medians is given.

2016 Czech-Polish-Slovak Match, 3

Tags: circumcircle , centroid , geometry

Let $ABC$ be an acute-angled triangle with $AB < AC$. Tangent to its circumcircle $\Omega$ at $A$ intersects the line $BC$ at $D$. Let $G$ be the centroid of $\triangle ABC$ and let $AG$ meet $\Omega$ again at $H \neq A$. Suppose the line $DG$ intersects the lines $AB$ and $AC$ at $E$ and $F$, respectively. Prove that $\angle EHG = \angle GHF$.(Slovakia)