Olympiad problems

1996 Spain Mathematical Olympiad, 2

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

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 $ABC$.

2014 Chile National Olympiad, 2

Tags: geometry , parallelogram , centroid , area

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 $EFGH$.

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.

2019 Czech-Polish-Slovak Junior Match, 2

Tags: geometry , centroid , equal segments

Let $ABC$ be a triangle with centroid $T$. Denote by $M$ the midpoint of $BC$. Let $D$ be a point on the ray opposite to the ray $BA$ such that $AB = BD$. Similarly, let $E$ be a point on the ray opposite to the ray $CA$ such that $AC = CE$. The segments $T D$ and $T E$ intersect the side $BC$ in $P$ and $Q$, respectively. Show that the points $P, Q$ and $M$ split the segment $BC$ into four parts of equal length.

1998 Czech and Slovak Match, 5

Tags: maximum value , trigonometry , equal angles , centroid , geometry

In a triangle $ABC$, $T$ is the centroid and $ \angle TAB = \angle ACT$. Find the maximum possible value of $sin \angle CAT +sin \angle CBT$.

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.

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

2015 Sharygin Geometry Olympiad, 7

Tags: geometry , angle , orthocenter , centroid

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)

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 $\vartriangle ABC$.

2018 Iranian Geometry Olympiad, 4

Tags: 3-dimensional geometry , 3d geometry , polyhedron , centroid , geometry

We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$. In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$. Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.) Proposed by Mahdi Etesamifard - Morteza Saghafian

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 $ABC$.

2024 Turkey Team Selection Test, 5

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

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 $HG$.

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 $n.$

2009 Singapore Senior Math Olympiad, 1

Tags: geometry , centroid , triangle

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

2012 Sharygin Geometry Olympiad, 3

Tags: geometry , centroid , incenter , perpendicular

Let $M$ and $I$ be the centroid and the incenter of a scalene triangle $ABC$, and let $r$ be its inradius. Prove that $MI = r/3$ if and only if $MI$ is perpendicular to one of the sides of the triangle. (A.Karlyuchenko)

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)

2003 Oral Moscow Geometry Olympiad, 4

Tags: geometry , incenter , centroid , perpendicular

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

2021 Sharygin Geometry Olympiad, 8.3

Tags: locus , centroid , speed , geometry

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 $ZAB$.

2004 Regional Olympiad - Republic of Srpska, 2

Tags: geometry , centroid , parallel , incenter

Let $ABC$ be a triangle, $T$ its centroid and $S$ its incenter. Prove that the following conditions are equivalent: (1) line $TS$ is parallel to one side of triangle $ABC$, (2) one of the sides of triangle $ABC$ is equal to the half-sum of the other two sides.

1982 Bulgaria National Olympiad, Problem 6

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

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

2011 Sharygin Geometry Olympiad, 1

Tags: orthocenter , centroid , tangent circle , geometry

In triangle $ABC$ the midpoints of sides $AC, BC$, vertex $C$ and the centroid lie on the same circle. Prove that this circle touches the circle passing through $A, B$ and the orthocenter of triangle $ABC$.

2018 District Olympiad, 2

Tags: geometry , centroid , orthocenter

Consider a right-angled triangle $ABC$, $\angle A = 90^{\circ}$ and points $D$ and $E$ on the leg $AB$ such that $\angle ACD \equiv \angle DCE \equiv \angle ECB$. Prove that if $3\overrightarrow{AD} = 2\overrightarrow{DE}$ and $\overrightarrow{CD} + \overrightarrow{CE} = 2\overrightarrow{CM}$ then $\overrightarrow{AB} = 4\overrightarrow{AM}$.

2015 Saudi Arabia GMO TST, 3

Tags: geometry , symmetry , 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

1957 Moscow Mathematical Olympiad, 365

Tags: geometry , ratio , equilateral , centroid , 3d geometry , sphere , tetrahedron

(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.$$ (b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$