Found problems: 85
2018 Regional Competition For Advanced Students, 2
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]
2014 Nordic, 2
Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.
2006 Sharygin Geometry Olympiad, 16
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?
2015 India Regional MathematicaI Olympiad, 5
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.
2007 Dutch Mathematical Olympiad, 1
Consider the equilateral triangle $ABC$ with $|BC| = |CA| = |AB| = 1$.
On the extension of side $BC$, we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$. Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$, and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$.
Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$.
[asy]
unitsize(1.5 cm);
pair[] A, B, C;
A[0] = (0,0);
B[0] = (1,0);
C[0] = dir(60);
A[1] = B[0] + dir(-60);
A[2] = C[0] + dir(120);
B[1] = C[0] + dir(60);
B[2] = A[0] + dir(240);
C[1] = A[0] + (-1,0);
C[2] = B[0] + (1,0);
draw(A[1]--A[2]);
draw(B[1]--B[2]);
draw(C[1]--C[2]);
draw(circumcircle(A[2],B[1],C[2]));
dot("$A$", A[0], SE);
dot("$A_1$", A[1], SE);
dot("$A_2$", A[2], NW);
dot("$B$", B[0], SW);
dot("$B_1$", B[1], NE);
dot("$B_2$", B[2], SW);
dot("$C$", C[0], N);
dot("$C_1$", C[1], W);
dot("$C_2$", C[2], E);
[/asy]
2023 Brazil National Olympiad, 2
Let $ABC$ be a right triangle in $B$, with height $BT$, $T$ on the hypotenuse $AC$. Construct the equilateral triangles $BTX$ and $BTY$ so that $X$ is in the same half-plane as $A$ with respect to $BT$ and $Y$ is in the same half-plane as $C$ with respect to $BT$. Point $P$ is the intersection of $AY$ and $CX$. Show that $$PA \cdot BC = PB \cdot CA = PC \cdot AB.$$
2015 Sharygin Geometry Olympiad, P4
In a parallelogram $ABCD$ the trisectors of angles $A$ and $B$ are drawn. Let $O$ be the common points of the trisectors nearest to $AB$. Let $AO$ meet the second trisector of angle $B$ at point $A_1$, and let $BO$ meet the second trisector of angle $A$ at point $B_1$. Let $M$ be the midpoint of $A_1B_1$. Line $MO$ meets $AB$ at point $N$ Prove that triangle $A_1B_1N$ is equilateral.
2015 Dutch IMO TST, 3
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.
2007 Sharygin Geometry Olympiad, 1
A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
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.
2007 Singapore Senior Math Olympiad, 3
In the equilateral triangle $ABC, M, N$ are the midpoints of the sides $AB, AC$, respectively. The line $MN$ intersects the circumcircle of $\vartriangle ABC$ at $K$ and $L$ and the lines $CK$ and $CL$ meet the line $AB$ at $P$ and $Q$, respectively. Prove that $PA^2 \cdot QB = QA^2 \cdot PB$.
2024 Bulgarian Autumn Math Competition, 8.4
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) $p_1+2p_2$
2023 Grosman Mathematical Olympiad, 7
The plane is colored with two colors so that the following property holds: for each real $a>0$ there is an equilateral triangle of side length $a$ whose $3$ vertices are of the same color.
Prove that for any three numbers $a,b,c>0$ for which the sum of any two is greater than the third there is a triangle with sides $a$, $b$, and $c$ whose $3$ vertices are of the same color.
2002 Rioplatense Mathematical Olympiad, Level 3, 3
Let $ABC$ be a triangle with $\angle C=60^o$. The point $P$ is the symmetric of $A$ with respect to the point of tangency of the circle inscribed with the side $BC$ . Show that if the perpendicular bisector of the $CP$ segment intersects the line containing the angle - bisector of $\angle B$ at the point $Q$, then the triangle $CPQ$ is equilateral.
2009 May Olympiad, 2
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 $\angle CED$.
2019 Romanian Master of Mathematics Shortlist, original P4
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 $ABC.$
2019 Brazil Team Selection Test, 1
Let $ABC$ be an acute triangle, with $\angle A > 60^\circ$, and let $H$ be it's orthocenter. Let $M$ and $N$ be points on $AB$ and $AC$, respectively, such that $\angle HMB = \angle HNC = 60^\circ$. Also, let $O$ be the circuncenter of $HMN$ and $D$ be a point on the semiplane determined by $BC$ that contains $A$ in such a way that $DBC$ is equilateral. Prove that $H$, $O$ and $D$ are collinear.
2023 IMO, 6
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]
2018 Irish Math Olympiad, 8
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 $\angle BFM = \angle BME$.
2009 Bosnia And Herzegovina - Regional Olympiad, 2
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
2021 Bolivian Cono Sur TST, 1
Inside a rhombus $ABCD$ with $\angle BAD=60$, points $F,H,G$ are choosen on lines $AD,DC,AC$ respectivily such that $DFGH$ is a paralelogram. Show that $BFH$ is a equilateral triangle.
2007 Sharygin Geometry Olympiad, 10
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.
2011 Israel National Olympiad, 4
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.
May Olympiad L1 - geometry, 2005.4
There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.
2018 Yasinsky Geometry Olympiad, 1
Points $A, B$ and $C$ lie on the same line so that $CA = AB$. Square $ABDE$ and the equilateral triangle $CFA$, are constructed on the same side of line $CB$. Find the acute angle between straight lines $CE$ and $BF$.