Olympiad problems

1940 Moscow Mathematical Olympiad, 063

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 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 $\angle XNM = \angle BAN$.

Denmark (Mohr) - geometry, 2010.5

Tags: trisector , geometry , semicircle , Equilateral

An equilateral triangle $ABC$ is given. With $BC$ as diameter, a semicircle is drawn outside the triangle. On the semicircle, points $D$ and $E$ are chosen such that the arc lengths $BD, DE$ and $EC$ are equal. Prove that the line segments $AD$ and $AE$ divide the side $BC$ into three equal parts. [img]https://1.bp.blogspot.com/-hQQV-Of96Ls/XzXCZjCledI/AAAAAAAAMV0/SwXa4mtEEm04onYbFGZiTc5NSpkoyvJLwCLcBGAsYHQ/s0/2010%2BMohr%2Bp5.png[/img]

2019 New Zealand MO, 5

Tags: combinatorial geometry , geometry , combinatorics , Equilateral , equilaterals

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

2023 Chile National Olympiad, 3

Tags: geometry , Equilateral , areas

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. $1011$ points $P_1$, $P_2$, $P_3$, $...$, $P_{1011}$ on the side $AC$ and $1011$ points $Q_1$, $Q_2$, $Q_3$, $...$ ,$ Q_{1011}$ on side AB (see figure) in such a way as to generate $2023$ triangles of equal area. Find the length of the segment $AP_{1011}$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/fea495c16a0b626e0c3882df66d66011a1a3af.png[/img] PS. Harder version of [url=https://artofproblemsolving.com/community/c4h3323135p30741470]2023 Chile NMO L1 P3[/url]

1975 Dutch Mathematical Olympiad, 4

Tags: geometry , analytic geometry , lattice points , Equilateral

Given is a rectangular plane coordinate system. (a) Prove that it is impossible to find an equilateral triangle whose vertices have integer coordinates. (b) In the plane the vertices $A, B$ and $C$ lie with integer coordinates in such a way that $AB = AC$. Prove that $\frac{d(A,BC)}{BC}$ is rational.

1990 Greece National Olympiad, 3

Tags: geometry , Equilateral

In a triangle $ABC$ with medians $AD$ and $BE$ , holds that $\angle CAD= \angle CBE=30^o$. Prove that triangle $ABC$ is equilateral.

Novosibirsk Oral Geo Oly VIII, 2019.1

Tags: geometry , distance , Equilateral

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]

2000 German National Olympiad, 3

Tags: Equilateral , angles , geometry

Suppose that an interior point $O$ of a triangle $ABC$ is such that the angles $\angle BAO,\angle CBO, \angle ACO$ are all greater than or equal to $30^o$. Prove that the triangle $ABC$ is equilateral.

1981 Czech and Slovak Olympiad III A, 3

Tags: geometry , Locus , square , Equilateral , Triangle

Let $ABCD$ be a unit square. Consider an equilateral triangle $XYZ$ with $X,Y$ as (inner or boundary) points of the square. Determine the locus $M$ of vertices $Z$ of all these triangles $XYZ$ and compute the area of $M.$

Cono Sur Shortlist - geometry, 1993.6

Tags: geometry , Equilateral , equal segments

Consider in the interior of an equilateral triangle $ABC$ points $D, E$ and $F$ such that$ D$ belongs to segment $BE$, $E$ belongs to segment $CF$ and$ F$ to segment $AD$. If $AD=BE = CF$ then $DEF$ is equilateral.

2017 BMT Spring, 13

Tags: geometry , Equilateral , square , area

$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?

2007 Sharygin Geometry Olympiad, 10

Tags: geometry , Locus , Equilateral , Equilateral Triangle , Locus problems

Find the locus of centers of regular triangles such that three given points $A, B, C$ lie respectively on three lines containing sides of the triangle.

1989 Austrian-Polish Competition, 2

Tags: Coloring , combinatorics , combinatorial geometry , Equilateral

Each point of the plane is colored by one of the two colors. Show that there exists an equilateral triangle with monochromatic vertices.

1987 Tournament Of Towns, (162) 6

Tags: equilaterals , Equilateral , Coloring , combinatorics

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?

2009 District Olympiad, 4

Tags: angles , geometry , isosceles , Equilateral

Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively, such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and $\angle CMP = 90 ^o$. a) Show that $\vartriangle NMB$ is isosceles. b) Determine $\angle CBM$.

1985 Tournament Of Towns, (080) T1

Tags: geometry , concurrent , Equilateral , angle bisector , median , altitude

A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .

III Soros Olympiad 1996 - 97 (Russia), 9.7

Tags: geometry , Equilateral , inscribed

Find the side of the smallest regular triangle that can be inscribed in a right triangle with an acute angle of $30^o$ and a hypotenuse of $2$. (All vertices of the required regular triangle must be located on different sides of this right triangle.)

Estonia Open Senior - geometry, 2002.2.3

Tags: geometry , parallelogram , rhombus , Equilateral

Let $ABCD$ be a rhombus with $\angle DAB = 60^o$. Let $K, L$ be points on its sides $AD$ and $DC$ and $M$ a point on the diagonal $AC$ such that $KDLM$ is a parallelogram. Prove that triangle $BKL$ is equilateral.

1977 Spain Mathematical Olympiad, 8

Tags: geometry , complex numbers , Equilateral

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.

1996 Estonia National Olympiad, 3

Tags: geometry , rotation , area , Equilateral

An equilateral triangle of side$ 1$ is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.

2020 Dutch BxMO TST, 4

Tags: equal segments , Equilateral , geometry

Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$. Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$.