Olympiad problems

2020 Israel National Olympiad, 3

In a convex hexagon $ABCDEF$ the triangles $BDF, ACE$ are equilateral and congruent. Prove that the three lines connecting the midpoints of opposite sides are concurrent.

1978 IMO Longlists, 4

Tags: equilateral triangle , symmetry , geometry , similar triangles

Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A' \neq S$ and $B' \neq S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.

2019 Romanian Master of Mathematics Shortlist, original P4

Tags: equilateral triangle , circumcircle , geometry

Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$

2015 IFYM, Sozopol, 7

Tags: equilateral triangle , geometry

In a square with side 1 are placed $n$ equilateral triangles (without having any parts outside the square) each with side greater than $\sqrt{\frac{2}{3}}$. Prove that all of the $n$ equilateral triangles have a common inner point.

2014 IFYM, Sozopol, 7

Tags: equilateral triangle , symmedian , geometry

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?

2005 Sharygin Geometry Olympiad, 7

Tags: geometry , concentric circles , angle , equilateral , equilateral triangle

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?

2018 Regional Competition For Advanced Students, 2

Tags: geometry , equilateral triangle , chord , perpendicular bisector

Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral. [i]Proposed by Stefan Leopoldseder[/i]

2009 May Olympiad, 2

Tags: geometry , angle chasing , equilateral triangle , convex quadrilateral , isosceles

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

2015 India Regional MathematicaI Olympiad, 5

Tags: geometry , equilateral triangle , incenter , circumcircle

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I.$ Let the internal angle bisectors of $\angle A,\angle B,\angle C$ meet $\Gamma$ in $A',B',C'$ respectively. Let $B'C'$ intersect $AA'$ at $P,$ and $AC$ in $Q.$ Let $BB'$ intersect $AC$ in $R.$ Suppose the quadrilateral $PIRQ$ is a kite; that is, $IP=IR$ and $QP=QR.$ Prove that $ABC$ is an equilateral triangle.

2005 Peru MO (ONEM), 3

Tags: equilateral triangle , equilateral , equal segments , angle , geometry

Let $A,B,C,D$, be four different points on a line $\ell$, so that $AB=BC=CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$.

2007 Singapore Junior Math Olympiad, 2

Tags: equilateral triangle , equilateral , parallelogram , geometry

Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.

2011 Israel National Olympiad, 4

Tags: geometry , equilateral triangle , tangent circle

Let $\alpha_1,\alpha_2,\alpha_3$ be three congruent circles that are tangent to each other. A third circle $\beta$ is tangent to them at points $A_1,A_2,A_3$ respectively. Let $P$ be a point on $\beta$ which is different from $A_1,A_2,A_3$. For $i=1,2,3$, let $B_i$ be the second intersection point of the line $PA_i$ with circle $\alpha_i$. Prove that $\Delta B_1B_2B_3$ is equilateral.

2014 Oral Moscow Geometry Olympiad, 5

Tags: geometry , area , equilateral , equilateral triangle , circumcircle , area of a triangle

Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

2023 ISL, G8

Tags: coaxial circles , equilateral triangle , geometry

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and $$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$ Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points. (Note: a scalene triangle is one where no two sides have equal length.) [i]Proposed by Ankan Bhattacharya, USA[/i]

2019 Pan-African Shortlist, G4

Tags: circumcenter , equilateral triangle , circumcircle , geometry

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral 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?

2009 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , equilateral triangle , fixed

Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant

VMEO IV 2015, 11.2

Tags: equilateral triangle , angle , geometry , isosceles

Given an isosceles triangle $BAC$ with vertex angle $\angle BAC =20^o$. Construct an equilateral triangle $BDC$ such that $D,A$ are on the same side wrt $BC$. Construct an isosceles triangle $DEB$ with vertex angle $\angle EDB = 80^o$ and $C,E$ are on the different sides wrt $DB$. Prove that the triangle $AEC$ is isosceles at $E$.

2018 Oral Moscow Geometry Olympiad, 6

Tags: equilateral triangle , equilateral , geometry , combinatorial geometry

Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.

2014 May Olympiad, 4

Tags: right triangle , isosceles triangle , equilateral triangle , angle , geometry

Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$

2007 Sharygin Geometry Olympiad, 10

Tags: locus , equilateral , equilateral triangle , locus problems , geometry

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.

1983 Brazil National Olympiad, 2

Tags: pyramid , geometry , equilateral triangle , 3d geometry , hexagon

An equilateral triangle $ABC$ has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side $a$ is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done, the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.

1998 May Olympiad, 2

Tags: equilateral triangle , area , geometry

Let $ABC$ be an equilateral triangle. $N$ is a point on the side $AC$ such that $\vec{AC} = 7\vec{AN}$, $M$ is a point on the side $AB$ such that $MN$ is parallel to $BC$ and $P$ is a point on the side $BC$ such that $MP$ is parallel to $AC$. Find the ratio of areas $\frac{ (MNP)}{(ABC)}$

2018 Irish Math Olympiad, 8

Tags: equilateral triangle , geometry , equal angles

Let $M$ be the midpoint of side $BC$ of an equilateral triangle $ABC$. The point $D$ is on $CA$ extended such that $A$ is between $D$ and $C$. The point $E$ is on $AB$ extended such that $B$ is between $A$ and $E$, and $|MD| = |ME|$. The point $F$ is the intersection of $MD$ and $AB$. Prove that $\angle BFM = \angle BME$.

2024 Bulgarian Autumn Math Competition, 8.4

Tags: lattice points , equilateral triangle , combinatorics , function

Let $n$ be a positive integers. Equilateral triangle with sides of length $n$ is split into equilateral triangles with side lengths $1$, forming a triangular lattice. Call an equilateral triangle with vertices in the lattice "important". Let $p_k$ be the number of unordered pairs of vertices in the lattice which participate in exactly $k$ important triangles. Find (as a function of $n$) (a) $p_0+p_1+p_2$ (b) $p_1+2p_2$