Found problems: 287
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 $AB, OC$.
Kyiv City MO Juniors Round2 2010+ geometry, 2012.8.5
In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$.
(Rozhkova Maria)
1999 Israel Grosman Mathematical Olympiad, 3
For every triangle $ABC$, denote by $D(ABC)$ the triangle whose vertices are the tangency points of the incircle of $\vartriangle ABC$ with the sides. Assume that $\vartriangle ABC$ is not equilateral.
(a) Prove that $D(ABC)$ is also not equilateral.
(b) Find in the sequence $T_1 = \vartriangle ABC, T_{k+1} = D(T_k)$ for $k \in N$ a triangle whose largest angle $\alpha$ satisfies $0 < \alpha -60^o < 0.0001^o$
2018 Romania National Olympiad, 3
On the sides $[AB]$ and $[BC]$ of the parallelogram $ABCD$ are constructed the equilateral triangles $ABE$ and $BCF,$ so that the points $D$ and $E$ are on the same side of the line $AB$, and $F$ and $D$ on different sides of the line $BC$. If the points $E,D$ and $F$ are collinear, then prove that $ABCD$ is rhombus.
2003 Czech And Slovak Olympiad III A, 2
On sides $BC,CA,AB$ of a triangle $ABC$ points $D,E,F$ respectively are chosen so that $AD,BE,CF$ have a common point, say $G$. Suppose that one can inscribe circles in the quadrilaterals $AFGE,BDGF,CEGD$ so that each two of them have a common point. Prove that triangle $ABC$ is equilateral.
1998 May Olympiad, 4
$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle.
[img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]
2015 Costa Rica - Final Round, G1
Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.
2007 Singapore Senior Math Olympiad, 3
In the equilateral triangle $ABC, M, N$ are the midpoints of the sides $AB, AC$, respectively. The line $MN$ intersects the circumcircle of $\vartriangle ABC$ at $K$ and $L$ and the lines $CK$ and $CL$ meet the line $AB$ at $P$ and $Q$, respectively. Prove that $PA^2 \cdot QB = QA^2 \cdot PB$.
1994 Czech And Slovak Olympiad IIIA, 5
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?
1999 Estonia National Olympiad, 5
Let $C$ be an interior point of line segment $AB$. Equilateral triangles $ADC$ and $CEB$ are constructed to the same side from $AB$. Find all points which can be the midpoint of the segment $DE$.
2018 Hanoi Open Mathematics Competitions, 2
What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$?
A. $3\sqrt7$ B. $\frac{3 \sqrt3}{2}$ C. $2\sqrt5$ D. $\frac{3\sqrt3}{8}$ E. $3\sqrt5$
1985 Greece National Olympiad, 2
a) Prove that a convex $n$-gon cannot have more than $3$ interior angles acute.
b) Prove that a convex $n$-gon that has $3$ interior angles equal to $60^0,$ is equilateral.
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.
1999 Ukraine Team Selection Test, 11
Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$
2023 Chile Junior Math Olympiad, 3
Let $\vartriangle ABC$ be an equilateral triangle with side $1$. Four points are marked $P_1$, $P_2$, $P_3$, $P_4$ on side $AC$ and four points $Q_1$, $Q_2$, $Q_3$, $Q_4$ on side $AB$ (see figure) of such a way to generate $9$ triangles of equal area. Find the length of segment $AP_4$.
[img]https://cdn.artofproblemsolving.com/attachments/5/f/29243932262cb963b376244f4c981b1afe87c6.png[/img]
PS. Easier version of [url=https://artofproblemsolving.com/community/c6h3323141p30741525]2023 Chile NMO L2 P3[/url]
OIFMAT I 2010, 2
In an acute angle $ \vartriangle ABC $, let $ AD, BE, CF $ be their altitudes (with $ D, E, F $ lying on $ BC, CA, AB $, respectively). Let's call $ O, H $ the circumcenter and orthocenter of $ \vartriangle ABC $, respectively. Let $ P = CF \cap AO $. Suppose the following two conditions are true:
$\bullet$ $ FP = EH $
$\bullet$ There is a circle that passes through points $ A, O, H, C $
Prove that the $ \vartriangle ABC $ is equilateral.
1974 Spain Mathematical Olympiad, 4
All three sides of an equilateral triangle are assumed to be reflective (except in the vertices), in such a way that they reflect the rays of light located in their plane, that fall on them and that come out of an interior point of the triangle.
Determine the path of a ray of light that, starting from a vertex of the triangle reach another vertex of the same after reflecting successively on the three sides. Calculate the length of the path followed by the light assuming that the side of the triangle measures $1$ m.
2015 Oral Moscow Geometry Olympiad, 2
The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$.
[img]https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png[/img]
1973 Chisinau City MO, 68
Inside the triangle $ABC$, point $O$ was chosen so that the triangles $AOB, BOC, COA$ turned out to be similar. Prove that triangle $ABC$ is equilateral.
1994 Tournament Of Towns, (426) 3
Two-mutually perpendicular lines $\ell$ and $m$ intersect each other at a point of the circumference of a circle, dividing it into three arcs. A point $M_i$ ($i = 1$,$2$,$3$) is taken on each arc so that the tangent line to the circumference at the point $M_i$ intersects $\ell$ and $m$ in two points at the same distance from $M_i$ (that is $M_i$ is the midpoint of the segment between them). Prove that the triangle $M_1M_2M_3$ is equilateral.
(Przhevalsky)
1969 Kurschak Competition, 2
A triangle has side lengths $a, b, c$ and angles $A, B, C$ as usual (with $b$ opposite $B$ etc). Show that if $$a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0$$ then the triangle is 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 $AB$)
Indonesia Regional MO OSP SMA - geometry, 2015.3
Given the isosceles triangle $ABC$, where $AB = AC$. Let $D$ be a point in the segment $BC$ so that $BD = 2DC$. Suppose also that point $P$ lies on the segment $AD$ such that: $\angle BAC = \angle BP D$. Prove that $\angle BAC = 2\angle DP C$.
Estonia Open Senior - geometry, 1993.5
Within an equilateral triangle $ABC$, take any point $P$. Let $L, M, N$ be the projections of $P$ on sides $AB, BC, CA$ respectively. Prove that $\frac{AP}{NL}=\frac{BP}{LM}=\frac{CP}{MN}$.
1967 German National Olympiad, 1
In a plane, a square $ABCD$ and a point $P$ located inside it are given. Let a point $ Q$ pass through all sides of the square. Describe the set of all those points $R$ in for which the triangle $PQR$ is equilateral.