Olympiad problems

2012 IFYM, Sozopol, 6

Let $A_1 B_1 C_1$ and $A_2 B_2 C_2$ be two oppositely oriented concentric equilateral triangles. Prove that the lines $A_1 A_2$ , $B_1 B_2$ , and $C_1 C_2$ intersect in one point.

2021 India National Olympiad, 5

Tags: geometry , angles , Equilateral Triangle , equal lengths , INMO , trigonometry

In a convex quadrilateral $ABCD$, $\angle ABD=30^\circ$, $\angle BCA=75^\circ$, $\angle ACD=25^\circ$ and $CD=CB$. Extend $CB$ to meet the circumcircle of triangle $DAC$ at $E$. Prove that $CE=BD$. [i]Proposed by BJ Venkatachala[/i]

2014 IFYM, Sozopol, 7

Tags: geometry , Symmedians , Equilateral Triangle

If $AG_a,BG_b$, and $CG_c$ are symmedians in $\Delta ABC$ ($G_a\in BC,G_b\in AC,G_c\in AB$), is it possible for $\Delta G_a G_b G_c$ to be equilateral when $\Delta ABC$ is not equilateral?

1995 Spain Mathematical Olympiad, 6

Tags: geometry , Equilateral Triangle

Let $C$ be a variable interior point of a fixed segment $AB$. Equilateral triangles $ACB' $ and $CBA'$ are constructed on the same side and $ABC' $ on the other side of the line $AB$. (a) Prove that the lines $AA' ,BB'$ , and $CC'$ meet at some point $P$. (b) Find the locus of $P$ as $C$ varies. (c) Prove that the centers $A'' ,B'' ,C''$ of the three triangles form an equilateral triangle. (d) Prove that $A'' ,B'',C''$ , and $P$ lie on a circle.

2015 Dutch IMO TST, 3

Tags: geometry , Equilateral , Equilateral Triangle

An equilateral triangle $ABC$ is given. On the line through $B$ parallel to $AC$ there is a point $D$, such that $D$ and $C$ are on the same side of the line $AB$. The perpendicular bisector of $CD$ intersects the line $AB$ in $E$. Prove that triangle $CDE$ is equilateral.

2014 Contests, 2

Tags: geometry , Equilateral Triangle , center of mass

Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered [u]clockwise[/u]. It is given that $A_3=B_3=C_3$. Let $M$ be the center of mass of $\Delta A_1B_1C_1$, and let $N$ be the center of mass of $\Delta A_2B_2C_2$. Prove that $\Delta A_3MN$ is an equilateral triangle.

2002 Tuymaada Olympiad, 2

Tags: geometry , ratio , Equilateral , Equilateral Triangle

Points on the sides $ BC $, $ CA $ and $ AB $ of the triangle $ ABC $ are respectively $ A_1 $, $ B_1 $ and $ C_1 $ such that $ AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1 $. Prove that if triangle $ A_1B_1C_1 $ is equilateral, then triangle $ ABC $ is also equilateral.

2005 Sharygin Geometry Olympiad, 9

Tags: Equilateral , Equilateral Triangle , perpendicular , midpoint , Centroid , geometry

Let $O$ be the center of a regular triangle $ABC$. From an arbitrary point $P$ of the plane, the perpendiculars were drawn on the sides of the triangle. Let $M$ denote the intersection point of the medians of the triangle , having vertices the feet of the perpendiculars. Prove that $M$ is the midpoint of the segment $PO$.

2005 JBMO Shortlist, 5

Tags: geometry , maximum , concentric , Equilateral Triangle

Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.

2005 Sharygin Geometry Olympiad, 7

Tags: Equilateral , Equilateral Triangle , concentric circles , geometry , angles

Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?