Olympiad problems

2013 Tournament of Towns, 3

Let $ABC$ be an equilateral triangle with centre $O$. A line through $C$ meets the circumcircle of triangle $AOB$ at points $D$ and $E$. Prove that points $A, O$ and the midpoints of segments $BD, BE$ are concyclic.

2012 Bundeswettbewerb Mathematik, 3

Tags: geometry , isosceles , Equilateral Triangle , Equilateral

An equilateral triangle $DCE$ is placed outside a square $ABCD$. The center of this triangle is denoted as $M$ and the intersection of the straight line $AC$ and $BE$ with $S$. Prove that the triangle $CMS$ is isosceles.

1971 Czech and Slovak Olympiad III A, 5

Tags: geometry , Equilateral , Triangles , Locus

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 $AC.$

2010 Denmark MO - Mohr Contest, 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]

2004 Paraguay Mathematical Olympiad, 3

Tags: geometry , Equilateral

In an equilateral triangle $ABC$, whose side is $4$, the line perpendicular to $AB$ is drawn through the point $ A$, the line perpendicular to $BC$ through point $ B$ and the line perpendicular to $CA$ through point $C$. These three lines determine another triangle. Calculate the perimeter of this triangle

Indonesia MO Shortlist - geometry, g4

Tags: geometry , Equilateral , triangle inequality

Inside the equilateral triangle $ABC$ lies the point $T$. Prove that $TA$, $TB$ and $TC$ are the lengths of the sides of a triangle.

2020-21 KVS IOQM India, 14

Tags: geometry , square , Equilateral

Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB, Q, R$ on $BC$ and $S$ on $AC$. If the area of the square $PQRS$ is $m +n\sqrt{k}$ where $m, n$ are integers and $k$ is a prime number then determine the value of $\sqrt{\frac{m+n}{k^2}}$.

2009 Greece Junior Math Olympiad, 2

Tags: Angle Chasing , Equilateral

From vertex $A$ of an equilateral triangle $ABC$, a ray $Ax$ intersects $BC$ at point $D$. Let $E$ be a point on $Ax$ such that $BA =BE$. Calculate $\angle AEC$.

2010 Greece Junior Math Olympiad, 2

Tags: geometry , rectangle , Equilateral , equal segments , perpendicular

Let $ABCD$ be a rectangle with sides $AB=a$ and $BC=b$. Let $O$ be the intersection point of it's diagonals. Extent side $BA$ towards $A$ at a segment $AE=AO$, and diagonal $DB$ towards $B$ at a segment $BZ=BO$. If the triangle $EZC$ is an equilateral, then prove that: i) $b=a\sqrt3$ ii) $AZ=EO$ iii) $EO \perp ZD$

Kyiv City MO 1984-93 - geometry, 1988.7.1

Tags: geometry , trapezoid , Equilateral , angles

An isosceles trapezoid is divided by each diagonal into two isosceles triangles. Determine the angles of the trapezoid.

2007 Hanoi Open Mathematics Competitions, 8

Tags: geometry , Locus , Equilateral

Let $ABC$ be an equilateral triangle. For a point $M$ inside $\vartriangle ABC$, let $D,E,F$ be the feet of the perpendiculars from $M$ onto $BC,CA,AB$, respectively. Find the locus of all such points $M$ for which $\angle FDE$ is a right angle.

2009 Cuba MO, 2

Tags: Equilateral , incenter , equal circles , geometry

Let $I$ be the incenter of an acute riangle $ABC$. Let $C_A(A, AI)$ be the circle with center $A$ and radius $AI$. Circles $C_B(B, BI)$, $C_C(C, CI) $ are defined in an analogous way. Let $X, Y, Z$ be the intersection points of $C_B$ with $C_C$, $C_C$ with $C_A$, $C_A$ with $C_B$ respectively (different than $I$) . Show that if the radius of the circle that passes through the points $X, Y, Z$ is equal to the radius of the circle that passes through points $A$, $B$ and $C$ then triangle $ABC$ is equilateral.

III Soros Olympiad 1996 - 97 (Russia), 10.7

Tags: geometry , Equilateral

An arbitrary point $M$ is taken inside a regular triangle $ABC$. Prove, that on sides $AB$, $BC$ and $CA$ one can choose points $C_1$, $A_1$ and $B_1$, respectively, so that $B_1C_1 = AM$, $C_1A_1 = BM$, $A_1B_1 = CM$. Find $BA$ if $AB_1= a$, $AC_1 = b$, $a>b$.

2004 Estonia National Olympiad, 3

Tags: vector , Equilateral , geometry

Let $K, L, M$ be the feet of the altitudes drawn from the vertices $A, B, C$ of triangle $ABC$, respectively. Prove that $\overrightarrow{AK} + \overrightarrow{BL} + \overrightarrow{CM} = \overrightarrow{O}$ if and only if $ABC$ is equilateral.

2001 Denmark MO - Mohr Contest, 5

Tags: geometry , Equilateral , square

Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

2013 Peru MO (ONEM), 3

Tags: geometry , angles , Angle Chasing , Equilateral

Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$ .

2020 BMT Fall, 9

Tags: geometry , circles , Equilateral

A circle $C$ with radius $3$ has an equilateral triangle inscribed in it. Let $D$ be a circle lying outside the equilateral triangle, tangent to $C$, and tangent to the equilateral triangle at the midpoint of one of its sides. The radius of $D$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

2021 Mediterranean Mathematics Olympiad, 3

Tags: geometry , incenter , Equilateral , Fixed point , fixed

Let $ABC$ be an equiangular triangle with circumcircle $\omega$. Let point $F\in AB$ and point $E\in AC$ so that $\angle ABE+\angle ACF=60^{\circ}$. The circumcircle of triangle $AFE$ intersects the circle $\omega$ in the point $D$. The halflines $DE$ and $DF$ intersect the line through $B$ and $C$ in the points $X$ and $Y$. Prove that the incenter of the triangle $DXY$ is independent of the choice of $E$ and $F$. (The angles in the problem statement are not directed. It is assumed that $E$ and $F$ are chosen in such a way that the halflines $DE$ and $DF$ indeed intersect the line through $B$ and $C$.)

Novosibirsk Oral Geo Oly VIII, 2016.6

Tags: geometry , equal segments , Equilateral

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 $AM, BM, CM$.

1987 Tournament Of Towns, (148) 5

Tags: geometry , Locus , perpendicular , Equilateral

Perpendiculars are drawn from an interior point $M$ of the equilateral triangle $ABC$ to its sides , intersecting them at points $D, E$ and $F$ . Find the locus of all points $M$ such that $DEF$ is a right triangle . (J . Tabov , Sofia)

Brazil L2 Finals (OBM) - geometry, 2012.4

Tags: pentagon , angles , Angle Chasing , region , Equilateral

The figure below shows a regular $ABCDE$ pentagon inscribed in an equilateral triangle $MNP$ . Determine the measure of the angle $CMD$. [img]http://4.bp.blogspot.com/-LLT7hB7QwiA/Xp9fXOsihLI/AAAAAAAAL14/5lPsjXeKfYwIr5DyRAKRy0TbrX_zx1xHQCK4BGAYYCw/s200/2012%2Bobm%2Bl2.png[/img]