Found problems: 24
2002 All-Russian Olympiad Regional Round, 8.4
Given a triangle $ABC$ with pairwise distinct sides. on his on the sides, regular triangles $ABC_1$, $BCA_1$, $CAB_1$. are constructed externally. Prove that triangle $A_1B_1C_1$ cannot be regular.
1966 Poland - Second Round, 5
Each of the sides $ BC, CA, AB $ of the triangle $ ABC $ was divided into three equal parts and on the middle sections of these sides as bases, equilateral triangles were built outside the triangle $ ABC $, the third vertices of which were marked with the letters $ A', B' , C' $ respectively. In addition, points $ A'', B'', C'' $ were determined, symmetrical to $ A', B', C' $ respectively with respect to the lines $ BC, CA, AB $. Prove that the triangles $ A'B'C' $ and $ A''B''C'' $ are equilateral and have the same center of gravity as the triangle $ ABC $.
2016 Ecuador NMO (OMEC), 3
Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.
1958 Poland - Second Round, 5
Outside triangle $ ABC $ equilateral triangles $ BMC $, $ CNA $, and $ APB $ are constructed. Prove that the centers $ S $, $ T $, $ U $ of these triangles form an equilateral triangle.
Novosibirsk Oral Geo Oly IX, 2022.6
Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.
2000 May Olympiad, 4
There are pieces in the shape of an equilateral triangle with sides $1, 2, 3, 4, 5$ and $6$ ($50$ pieces of each size). You want to build an equilateral triangle of side $7$ using some of these pieces, without gaps or overlaps. What is the least number of pieces needed?
1972 Dutch Mathematical Olympiad, 1
Prove that for every $n \in N$, $n > 6$, every equilateral triangle can be divided into $n$ pieces, which are also equilateral triangles.
2022 Novosibirsk Oral Olympiad in Geometry, 6
Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.
1946 Moscow Mathematical Olympiad, 107
Given points $A, B, C$ on a line, equilateral triangles $ABC_1$ and $BCA_1$ constructed on segments $AB$ and $BC$, and midpoints $M$ and $N$ of $AA_1$ and $CC_1$, respectively. Prove that $\vartriangle BMN$ is equilateral. (We assume that $B$ lies between $A$ and $C$, and points $A_1$ and $C_1$ lie on the same side of line $AB$)
2019 New Zealand MO, 5
An equilateral triangle is partitioned into smaller equilateral triangular pieces. Prove that two of the pieces are the same size.
2001 All-Russian Olympiad Regional Round, 9.4
The target is a triangle divided by three families of parallel lines into $100$ equal regular triangles with single sides. A sniper shoots at a target. He aims at triangle and hits either it or one of the sides adjacent to it. He sees the results of his shooting and can choose when stop shooting. What is the greatest number of triangles he can with a guarantee of hitting five times?
1987 Tournament Of Towns, (162) 6
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?
1990 Tournament Of Towns, (277) 2
A point $M$ is chosen on the arc $AC$ of the circumcircle of the equilateral triangle $ABC$. $P$ is the midpoint of this arc, $N$ is the midpoint of the chord $BM$ and $K$ is the foot of the perpendicular drawn from $P$ to $MC$. Prove that the triangle $ANK$ is equilateral.
(I Nagel, Yevpatoria)
2020 Malaysia IMONST 2, 2
Prove that for any integer $n\ge 6$ we can divide an equilateral triangle completely into $n$ smaller equilateral triangles.
1989 Tournament Of Towns, (229) 3
The plane is cut up into equilateral triangles by three families of parallel lines.
Is it possible to find $4$ vertices of these triangles which form a square?
2012 Austria Beginners' Competition, 4
A segment $AB$ is given. We erect the equilateral triangles $ABC$ and $ADB$ above and below $AB$. We denote the midpoints of $AC$ and $BC$ by $E$ and $F$ respectively. Prove that the straight lines $DE$ and $DF$ divide the segment $AB$ into three parts of equal length .
1984 All Soviet Union Mathematical Olympiad, 373
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.
1992 All Soviet Union Mathematical Olympiad, 578
An equilateral triangle side $10$ is divided into $100$ equilateral triangles of side $1$ by lines parallel to its sides. There are m equilateral tiles of $4$ unit triangles and $25 - m$ straight tiles of $4$ unit triangles (as shown below). For which values of $m$ can they be used to tile the original triangle. [The straight tiles may be turned over.]
1993 Tournament Of Towns, (395) 3
Consider the hexagon which is formed by the vertices of two equilateral triangles (not necessarily equal) when the triangles intersect. Prove that the area of the hexagon is unchanged when one of the triangles is translated (without rotating) relative to the other in such a way that the hexagon continues to be defined.
(V Proizvolov)
2020 Puerto Rico Team Selection Test, 1
We have $10,000$ identical equilateral triangles. Consider the largest regular hexagon that can be formed with these triangles without overlapping. How many triangles will not be used?
1981 All Soviet Union Mathematical Olympiad, 309
Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral
1990 All Soviet Union Mathematical Olympiad, 518
An equilateral triangle of side $n$ is divided into $n^2$ equilateral triangles of side $1$. A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least $n$ vertices.
2014 Costa Rica - Final Round, 4
Consider the isosceles triangle $ABC$ inscribed in the semicircle of radius $ r$. If the $\vartriangle BCD$ and $\vartriangle CAE$ are equilateral, determine the altitude of $\vartriangle DEC$ on the side $DE$ in terms of $ r$.
[img]https://cdn.artofproblemsolving.com/attachments/6/3/772ff9a1fd91e9fa7a7e45ef788eec7a1ba48e.png[/img]
Oliforum Contest V 2017, 2
Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$.
(Emanuele Tron)