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: 296

2019 BMT Spring, 7
Tags: geometry , equilateral , ratio , hexagon , area
Let $\vartriangle ABC$ be an equilateral triangle with side length $M$ such that points $E_1$ and $E_2$ lie on side $AB$, $F_1$ and $F_2$ lie on side $BC$, and $G1$ and $G2$ lie on side $AC$, such that $$m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2}$$ and the area of polygon $E_1E_2F_1F_2G_1G_2$ equals the combined areas of $\vartriangle AE_1G_2$, $\vartriangle BF_1E_2$, and $\vartriangle CG_1F_2$. Find the ratio $\frac{m}{M}$. [img]https://cdn.artofproblemsolving.com/attachments/a/0/88b36c6550c42d913cdddd4486a3dde251327b.png[/img]

2002 Junior Balkan Team Selection Tests - Romania, 3
Tags: combinatorics , combinatorial geometry , equilateral
A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines.

1977 Spain Mathematical Olympiad, 8
Tags: complex number , equilateral , geometry
Determine a necessary and sufficient condition for the affixes of three complex numbers $z_1$ , $z_2$ and $z_3$ are the vertices of an equilateral triangle.

1982 Bundeswettbewerb Mathematik, 2
Tags: triangle , equilateral , projection , space , geometry
Decide whether every triangle $ABC$ in space can be orthogonally projected onto a plane such that the projection is an equilateral triangle $A'B'C'$.

1984 All Soviet Union Mathematical Olympiad, 373
Tags: vector , geometry , equilateral
Given two equilateral triangles $A_1B_1C_1$ and $A_2B_2C_2$ in the plane. (The vertices are mentioned counterclockwise.) We draw vectors $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$, from the arbitrary point $O$, equal to $\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}$ respectively. Prove that the triangle $ABC$ is equilateral.

2023 Adygea Teachers' Geometry Olympiad, 4
Tags: geometry , equilateral
In the equilateral triangle $ABC$ ($AB = 2$), cevians are drawn that do not intersect at one point. It turned out that the pairwise intersection points of these cevians lie on the inscribed circle of triangle $ABC$. Find the length of the cevian segment from the vertex of the triangle to the nearest point of intersection with the circle.

1977 Swedish Mathematical Competition, 2
Tags: geometry , equilateral
There is a point inside an equilateral triangle side $d$ whose distance from the vertices is $3, 4, 5$. Find $d$.

1987 Tournament Of Towns, (162) 6
Tags: combinatorics , equilateral , coloring
An equilateral triangle is divided by lines, parallel to its sides, into equilateral triangles, all of the same size. One of the smaller triangles is black while the others are white. It is permitted to intersect simultaneously some small triangles with a line parallel to any side of the original triangle and to change the colour of each intersected small triangle from one colour to the other . Is it always possible to find a sequence of such operations so that the smaller triangles all become white?

2008 Postal Coaching, 6
Tags: combinatorial geometry , point , equilateral , combinatorics
A set of points in the plane is called [i]free [/i] if no three points of the set are the vertices of an equilateral triangle. Prove that any set of $n$ points in the plane has a free subset of at least $\sqrt{n}$ points

1963 All Russian Mathematical Olympiad, 032
Tags: geometry , equilateral , minimum
Given equilateral triangle with the side $l$. What is the minimal length $d$ of a brush (segment), that will paint all the triangle, if its ends are moving along the sides of the triangle.

1971 Czech and Slovak Olympiad III A, 5
Tags: triangle , equilateral , locus , geometry
Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$

2001 Denmark MO - Mohr Contest, 5
Tags: equilateral , square , geometry
Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

2007 Hanoi Open Mathematics Competitions, 7
Tags: combinatorial geometry , equilateral , geometry
Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side $4$. Prove that some three of these points are vertices of a triangle whose area is not greater than $\sqrt3$.

2018 Malaysia National Olympiad, A1
Tags: hexagon , equilateral , geometry
Hassan has a piece of paper in the shape of a hexagon. The interior angles are all $120^o$, and the side lengths are $1$, $2$, $3$, $4$, $5$, $6$, although not in that order. Initially, the paper is in the shape of an equilateral triangle, then Hassan has cut off three smaller equilateral triangle shapes, one at each corner of the paper. What is the minimum possible side length of the original triangle?

Novosibirsk Oral Geo Oly VIII, 2019.1
Tags: distance , equilateral , geometry
Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa. [url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]

2006 Bosnia and Herzegovina Junior BMO TST, 2
Tags: geometry , equilateral , altitude
In an acute triangle $ABC$, $\angle C = 60^o$. If $AA'$ and $BB'$ are two of the altitudes and $C_1$ is the midpoint of $AB$, prove that triangle $C_1A'B'$ is equilateral.

1966 Poland - Second Round, 6
Tags: geometry , fixed , equilateral
Prove that the sum of the squares of the right-angled projections of the sides of a triangle onto the line $ p $ of the plane of this triangle does not depend on the position of the line $ p $ if and only if it the triangle is equilateral.

2019 Saudi Arabia Pre-TST + Training Tests, 2.3
Tags: equal segments , geometry , equilateral , collinear
Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that i) $AX = BY = CZ$ ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order. Prove that $XY Z$ is an equilateral triangle.

1970 Dutch Mathematical Olympiad, 3
Tags: symmetry , equilateral , square , geometry
The points $P,Q,R$ and $A,B,C,D$ lie on a circle (clockwise) such that $\vartriangle PQR$ is equilateral and $ABCD$ is a square. The points $A$ and $P$ coincide. Prove that the symmetric of $B$ and $D$ wrt $PQ$ and $PR$ respectively lie on the sidelines of the symmetric square wrt $QR$.

Ukrainian From Tasks to Tasks - geometry, 2012.9
Tags: equilateral , geometry
In the triangle $ABC$, the angle $A$ is equal to $60^o$, and the median $BD$ is equal to the altitude $CH$. Prove that this triangle is equilateral.

2023 Yasinsky Geometry Olympiad, 6
Tags: equilateral , geometry
An acute triangle $ABC$ is surrounded by equilateral triangles $KLM$ and $PQR$ such that its vertices lie on the sides of these equilateral triangle as shown on the picture. Lines $PK$ and $QL$ intersect at point $D$. Prove that $\angle ABC + \angle PDQ = 120^o$. (Yurii Biletskyi) [img]https://cdn.artofproblemsolving.com/attachments/4/6/32d3f74f07ca6a8edcabe4a08aa321eb3a5010.png[/img]

2007 Sharygin Geometry Olympiad, 1
Tags: geometry , angle , equilateral , equilateral triangle , isosceles triangle , isosceles
A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.

2015 Oral Moscow Geometry Olympiad, 5
Tags: geometric inequality , geometry , rhombus , equilateral
On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.

1993 Austrian-Polish Competition, 9
Tags: incircle , geometry , equilateral triangle , equilateral , inradius
Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$. Denote by $a$ the side length of triangle $ABC$, by $r_1$ the inradius of triangle $PAC$, and by $r_2$ the exradius of triangle $PBC$ opposite $P$. Find the sum $r_1+r_2$ as a function in $a$.

2002 Tuymaada Olympiad, 2
Tags: geometry , equilateral , equilateral triangle , ratio
Points on the sides $ BC $, $ CA $ and $ AB $ of the triangle $ ABC $ are respectively $ A_1 $, $ B_1 $ and $ C_1 $ such that $ AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1 $. Prove that if triangle $ A_1B_1C_1 $ is equilateral, then triangle $ ABC $ is also equilateral.