This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 116

1992 IMO Longlists, 5
Tags: centroid , circumcircle , incenter , geometry
Let $I,H,O$ be the incenter, centroid, and circumcenter of the nonisosceles triangle $ABC$. Prove that $AI \parallel HO$ if and only if $\angle BAC =120^{\circ}$.

1969 IMO Longlists, 57
Tags: centroid , triangle , geometry
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 $ .

1996 Spain Mathematical Olympiad, 2
Tags: centroid , geometry , isosceles
Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles

2012 Tournament of Towns, 6
Tags: 3d geometry , centroid , sphere , icosahedron , geometry , combinatorial geometry
(a) A point $A$ is marked inside a sphere. Three perpendicular lines drawn through $A$ intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines. (b) An icosahedron with the centre $A$ is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from $A$ to the vertices of the icosahedron mark $12$ points on the sphere. Then the icosahedron is rotated about its centre. New rays mark new $12$ points on the sphere. Let $O$ and $N$ be the centres of mass of old and new points respectively. Prove that $O = N$.

2004 Junior Balkan Team Selection Tests - Romania, 2
Tags: equilateral , centroid , midpoint , geometry , cyclic quadrilateral
Let $M,N, P$ be the midpoints of the sides $BC,CA,AB$ of the triangle $ABC$, respectively, and let $G$ be the centroid of the triangle. Prove that if $BMGP$ is cyclic and $2BN = \sqrt3 AB$ , then triangle $ABC$ is equilateral.

1969 IMO Shortlist, 57
Tags: centroid , triangle , geometry
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 Ukraine Team Selection Test, 9
Tags: centroid , collinear , geometry , incircle
The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$, respectively. Let $S$ be the intersection point of lines passing through points $B$ and $C$ and parallel to $A_1C_1$ and $A_1B_1$ respectively, $A_0$ be the foot of the perpendicular drawn from point $A_1$ on $B_1C_1$, $G_1$ be the centroid of triangle $A_1B_1C_1$, $P$ be the intersection point of the ray $G_1A_0$ with $\omega$. Prove that points $S, A_1$, and $P$ lie on a straight line.

Durer Math Competition CD 1st Round - geometry, 2012.C5
Tags: centroid , geometry , incenter
In a triangle, the line between the center of the inscribed circle and the center of gravity is parallel to one of the sides. Prove that the sidelengths form an arithmetic sequence.

1955 Kurschak Competition, 3
Tags: analytic geometry , lattice , centroid , geometry
The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid.

2017 BMT Spring, 16
Tags: area , geometric inequality , centroid , area of a triangle , geometry
Let $ABC$ be a triangle with $AB = 3$, $BC = 5$, $AC = 7$, and let $ P$ be a point in its interior. If $G_A$, $G_B$, $G_C$ are the centroids of $\vartriangle PBC$, $\vartriangle PAC$, $\vartriangle PAB$, respectively, find the maximum possible area of $\vartriangle G_AG_BG_C$.

1981 Bundeswettbewerb Mathematik, 2
Tags: parallel , classic , centroid , geometry
Prove that if the sides $a, b, c$ of a non-equilateral triangle satisfy $a + b = 2c$, then the line passing through the incenter and centroid is parallel to one of the sides of the triangle.

2009 Singapore Senior Math Olympiad, 1
Tags: centroid , triangle , geometry
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 $ .

2006 Mathematics for Its Sake, 2
Tags: vector geometry , cevian , centroid , center of mass , 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]

1956 Moscow Mathematical Olympiad, 328
Tags: centroid , concurrency , midpoint , geometry
In a convex quadrilateral $ABCD$, consider quadrilateral $KLMN$ formed by the centers of mass of triangles $ABC, BCD, DBA, CDA$. Prove that the straight lines connecting the midpoints of the opposite sides of quadrilateral $ABCD$ meet at the same point as the straight lines connecting the midpoints of the opposite sides of $KLMN$.

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.)

2006 BAMO, 3
Tags: centroid , median , geometry
In triangle $ABC$, choose point $A_1$ on side $BC$, point $B_1$ on side $CA$, and point $C_1$ on side $AB$ in such a way that the three segments $AA_1, BB_1$, and $CC_1$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $A_1B_1C_1$. Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.

2012 Grigore Moisil Intercounty, 3
Tags: vector geometry , centroid , center of mass , geometry
Let $ M,N,P $ on the sides $ AB,BC,CA, $ respectively, of a triangle $ ABC $ such that $ AM=BN=CP $ and such that $$ AB\cdot \overrightarrow{AT} +BC\cdot \overrightarrow{BT} +CA\cdot \overrightarrow{CT} =0, $$ where $ T $ is the centroid of $ MNP. $ Prove that $ ABC $ is equilateral.

2023-24 IOQM India, 14
Tags: median , coordinate geometry , centroid , geometry
Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroid of the triangle $D E F$. Find the length $G K$.

Cono Sur Shortlist - geometry, 1993.14
Tags: centroid , geometric inequality , geometry , min
Prove that the sum of the squares of the distances from a point $P$ to the vertices of a triangle $ABC$ is minimum when $ P$ is the centroid of the triangle.

1974 Chisinau City MO, 83
Tags: centroid , equilateral , locus , geometry
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 $AB, OC$.

2004 Regional Olympiad - Republic of Srpska, 2
Tags: centroid , parallel , incenter , geometry
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.

1998 Czech and Slovak Match, 5
Tags: centroid , maximum value , equal angles , geometry , trigonometry
In a triangle $ABC$, $T$ is the centroid and $ \angle TAB = \angle ACT$. Find the maximum possible value of $sin \angle CAT +sin \angle CBT$.

1991 Tournament Of Towns, (318) 5
Tags: geometric transformation , centroid , equilateral , rotation , geometry
Let $M$ be a centre of gravity (the intersection point of the medians) of a triangle $ABC$. Under rotation by $120$ degrees about the point $M$, the point $B$ is taken to the point $P$; under rotation by $240$ degrees about $M$, the point $C$ is taken to the point $Q$. Prove that either $APQ$ is an equilateral triangle, or the points $A, P, Q$ coincide. (Bykovsky, Khabarovsksk)

1968 IMO Shortlist, 9
Tags: inequalities , centroid , area of a triangle , geometric inequality , geometry
Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality \[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\] Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

Estonia Open Junior - geometry, 2002.1.4
Tags: equal area , centroid , geometry
Consider a point $M$ inside triangle $ABC$ such that triangles $ABM, BCM$ and $CAM$ have equal areas. Prove that $M$ is the intersection point of the medians of triangle $ABC$.