Olympiad problems

1973 Spain Mathematical Olympiad, 6

An equilateral triangle of altitude $1$ is considered. For every point $P$ on the interior of the triangle, denote by $x, y , z$ the distances from the point $P$ to the sides of the triangle. a) Prove that for every point $P$ inside the triangle it is true that $x + y + z = 1$. b) For which points of the triangle does it hold that the distance to one side is greater than the sum of the distances to the other two? c) We have a bar of length $1$ and we break it into three pieces. find the probability that with these pieces a triangle can be formed.

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 $AB, OC$.

2020 BMT Fall, 11

Tags: equilateral , area , circles , geometry

Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p}-\frac{q\pi}{r}$, where $p, q$, and $ r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/f349a807583a83f93ba413bebf07e013265551.png[/img]

Kyiv City MO 1984-93 - geometry, 1993.9.3

Tags: equilateral , fixed , geometry

The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.

2012 Denmark MO - Mohr Contest, 5

Tags: equal segments , hexagon , equilateral , equal angles , geometry

In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.

Estonia Open Senior - geometry, 2018.1.1

Tags: equilateral , lattice points , geometry

Is there an equilateral triangle in the coordinate plane, both coordinates of each vertex of which are integers?

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)

1998 Italy TST, 2

Tags: equilateral , median , altitude , angle bisector , geometry

In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.

1993 Bundeswettbewerb Mathematik, 2

Tags: combinatorial geometry , equilateral , plane , geometry , combinatorics

Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$

1991 Swedish Mathematical Competition, 6

Tags: area , geometric inequalities , equilateral , geometry

Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

2006 Sharygin Geometry Olympiad, 16

Tags: equilateral , equilateral triangle , geometry

Regular triangles are built on the sides of the triangle $ABC$. It turned out that their vertices form a regular triangle. Is the original triangle regular also?

2016 Novosibirsk Oral Olympiad in Geometry, 6

Tags: equal segments , equilateral , geometry

An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

2015 Oral Moscow Geometry Olympiad, 2

Tags: angle , equilateral , geometry , square

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]

1997 Abels Math Contest (Norwegian MO), 2a

Tags: geometry , sum , independent , equilateral , perpendicular

Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.

2020 Yasinsky Geometry Olympiad, 2

Tags: equilateral , angle , square , geometry

An equilateral triangle $BDE$ is constructed on the diagonal $BD$ of the square $ABCD$, and the point $C$ is located inside the triangle $BDE$. Let $M$ be the midpoint of $BE$. Find the angle between the lines $MC$ and $DE$. (Dmitry Shvetsov)

2007 Peru MO (ONEM), 4

Tags: geometry , rhombus , equilateral , equal angles

Let $ABCD$ be rhombus $ABCD$ where the triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the intersection point of the diagonals $AC$ and $BD$. Prove that $\angle XMN = \angle\ DAM$ and $\angle XNM = \angle BAN$.

1981 All Soviet Union Mathematical Olympiad, 309

Tags: equilateral , geometry , vector

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

2021 Novosibirsk Oral Olympiad in Geometry, 6

Tags: equilateral , geometry , equal segments

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

2019 New Zealand MO, 5

Tags: equilateral , combinatorics , combinatorial geometry , geometry

An equilateral triangle is partitioned into smaller equilateral triangular pieces. Prove that two of the pieces are the same size.

Kyiv City MO 1984-93 - geometry, 1989.8.5

Tags: inscribed , equilateral , construction , geometry , right triangle

The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.

1967 German National Olympiad, 1

Tags: locus , equilateral , square , geometry

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.

1995 Bundeswettbewerb Mathematik, 2

Tags: locus , equilateral , geometry

A line $g$ and a point $A$ outside $g$ are given in a plane. A point $P$ moves along $g$. Find the locus of the third vertices of equilateral triangles whose two vertices are $A$ and $P$.

2009 Moldova National Olympiad, 9.3

Tags: equilateral , parallel , geometry

Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.

2005 Oral Moscow Geometry Olympiad, 5

Tags: circumcircle , equilateral , construction , geometry

The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle. (A. Zaslavsky)

1994 Tournament Of Towns, (426) 3

Tags: equilateral , geometry

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)