Olympiad problems

1974 Chisinau City MO, 83

Let $O$ be the center of the regular triangle $ABC$. Find the set of all points $M$ such that any line containing the point $M$ intersects one of the segments $AB, OC$.

2013 Saudi Arabia Pre-TST, 1.4

Tags: geometry , cyclic quadrilateral , centroid

$ABC$ 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 $AB \cdot CC' = AC \cdot BB'$:

2024 Romania National Olympiad, 2

Tags: geometry , circumcircle , pentagon , inscriptible , centroid

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]

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.

1989 Greece National Olympiad, 2

Tags: analytic geometry , centroid , combinatorial geometry , lattice points , weight , geometry

On the plane we consider $70$ points $A_1,A_2,...,A_{70}$ with integer coodinates. Suppose each pooints has weight $1$ and the centers of gravity of the triangles $ A_1A_2A_3$, $A_2A_3A_4$, $..$., $A_{68}A_{69}A_{70}$, $A_{69}A_{70}A_{1}$, $A_{70}A_{1}A_{2}$ have integer coodinates. Prove that the centers of gravity of any triple $A_i,A_j,...,A_{k}$ has integer coodinates.

2024 Turkey Team Selection Test, 5

Tags: geometry , olympiad geometry , orthocenter , centroid , complex number geometry

In a scalene triangle $ABC$, $H$ is the orthocenter, and $G$ is the centroid. Let $A_b$ and $A_c$ be points on $AB$ and $AC$, respectively, such that $B$, $C$, $A_b$, $A_c$ are cyclic, and the points $A_b$, $A_c$, $H$ are collinear. $O_a$ is the circumcenter of the triangle $AA_bA_c$. $O_b$ and $O_c$ are defined similarly. Prove that the centroid of the triangle $O_aO_bO_c$ lies on the line $HG$.

1936 Eotvos Mathematical Competition, 2

Tags: equal area , geometry , centroid

$S$ 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 $ABC$.

2021 Sharygin Geometry Olympiad, 8.3

Tags: locus , centroid , geometry , speed

Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$. Cockroach $A$ runs twice as slow than $B$, and thee times as slow than $C$. Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of centroids of triangles $ZAB$.

1982 Bulgaria National Olympiad, Problem 6

Tags: triangle , square , centroid , triangle centers , locus , geometry

Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.

1996 Spain Mathematical Olympiad, 2

Tags: isosceles , centroid , geometry

Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles

2009 Sharygin Geometry Olympiad, 6

Tags: geometry , centroid , incenter , parallel

Let $M, I$ be the centroid and the incenter of triangle $ABC, A_1$ and $B_1$ be the touching points of the incircle with sides $BC$ and $AC, G$ be the common point of lines $AA_1$ and $BB_1$. Prove that angle $\angle CGI$ is right if and only if $GM // AB$. (A.Zaslavsky)

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 $EF$.

2016 BMT Spring, 19

Tags: 3d geometry , tetrahedron , limit , centroid , geometry

Regular tetrahedron $P_1P_2P_3P_4$ has side length $1$. Define $P_i$ for $i > 4$ to be the centroid of tetrahedron $P_{i-1}P_{i-2}P_{i-3}P_{i-4}$, and $P_{ \infty} = \lim_{n\to \infty} P_n$. What is the length of $P_5P_{ \infty}$?

Kyiv City MO Seniors 2003+ geometry, 2004.10.5

Tags: geometry , centroid , orthocenter

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 $ABC$.

2017 Bundeswettbewerb Mathematik, 3

Tags: incenter , centroid , perpendicular , 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.

1999 Estonia National Olympiad, 3

Tags: euler line , centroid , orthocenter , parallel , trigonometry , geometry

Prove that the line segment, joining the orthocenter and the intersection point of the medians of the acute-angled triangle $ABC$ is parallel to the side $AB$ iff $\tan \angle A \cdot \tan \angle B = 3$.

1966 IMO Longlists, 55

Tags: geometry , centroid , locus , locus problems

Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$

1999 Spain Mathematical Olympiad, 5

Tags: inequalities , distance , geometric inequality , centroid , geometry

The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that: a) $g_a,g_b,g_c \ge \frac{2}{3}r$ b) $g_a+g_b+g_c \ge 3r$

Kyiv City MO Seniors Round2 2010+ geometry, 2019.11.3.1

Tags: geometry , incenter , centroid , equal segments

It is known that in the triangle $ABC$ the smallest side is $BC$. Let $X, Y, K$ and $L$ - points on the sides $AB, AC$ and on the rays $CB, BC$, respectively, are such that $BX = BK = BC =CY =CL$. The line $KX$ intersects the line $LY$ at the point $M$. Prove that the intersection point of the medians $\vartriangle KLM$ coincides with the center of the inscribed circle $\vartriangle ABC$.

2024 Nigerian MO Round 2, Problem 5

Tags: geometry , centroid

Let the centroid of the triangle $ABC$ be $G$ and let the line parallel to $\overline{BC}$ that passes through $A$ be $l$. Define a point, $D$ on $l$ such that $\angle DGC=90^o$. Prove that \[2[ADCG]\leq AB\cdot DC\] For clarification, [ADGC] represents the area of the quadrilateral ADGC.

2021 Yasinsky Geometry Olympiad, 4

Tags: centroid , circumradius , geometry

$K$ is an arbitrary point inside the acute-angled triangle $ABC$, in which $\angle A = 30^o$. $F$ and $N$ are the points of intersection of the medians in the triangles $AKC$ and $AKB$, respectively . It is known that $FN = q$. Find the radius of the circle circumscribed around the triangle $ABC$. (Grigory Filippovsky)

STEMS 2023 Math Cat A, 1

Tags: geometry , incenter , centroid

If in triangle $ABC$ , $AC$=$15$, $BC$=$13$ and $IG||AB$ where $I$ is the incentre and $G$ is the centroid , what is the area 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 $n.$

1979 Swedish Mathematical Competition, 6

Tags: geometric inequalities , geometry , centroid

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 $GA > GB > GC$.

2005 Sharygin Geometry Olympiad, 9

Tags: equilateral , equilateral triangle , perpendicular , midpoint , centroid , geometry

Let $O$ be the center of a regular triangle $ABC$. From an arbitrary point $P$ of the plane, the perpendiculars were drawn on the sides of the triangle. Let $M$ denote the intersection point of the medians of the triangle , having vertices the feet of the perpendiculars. Prove that $M$ is the midpoint of the segment $PO$.