Found problems: 86
1961 All-Soviet Union Olympiad, 4
Point $P$ and equilateral triangle $ABC$ satisfy $|AP|=2$, $|BP|=3$. Maximize $|CP|$.
2006 Sharygin Geometry Olympiad, 8.1
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
1986 IMO Shortlist, 17
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.
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.
1994 Nordic, 1
Let $O$ be an interior point in the equilateral triangle $ABC$, of side length $a$. The lines $AO, BO$, and $CO$ intersect the sides of the triangle in the points $A_1, B_1$, and $C_1$. Show that $OA_1 + OB_1 + OC_1 < a$.
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 $PA^2 \cdot QB = QA^2 \cdot PB$.
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.
2005 Sharygin Geometry Olympiad, 9
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 $PO$.
May Olympiad L2 - geometry, 2009.2
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 $\angle CED$.
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 Singapore Junior Math Olympiad, 2
Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.
2010 Sharygin Geometry Olympiad, 3
Points $X,Y,Z$ lies on a line (in indicated order). Triangles $XAB$, $YBC$, $ZCD$ are regular, the vertices of the first and the third triangle are oriented counterclockwise and the vertices of the second are opposite oriented. Prove that $AC$, $BD$ and $XY$ concur.
V.A.Yasinsky
2014 May Olympiad, 4
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 $\angle CAP$
2008 Junior Balkan Team Selection Tests - Moldova, 7
In an acute triangle $ABC$, points $A_1, B_1, C_1$ are the midpoints of the sides $BC, AC, AB$, respectively. It is known that $AA_1 = d(A_1, AB) + d(A_1, AC)$, $BB1 = d(B_1, AB) + d(A_1, BC)$, $CC_1 = d(C_1, AC) + d(C_1, BC)$, where $d(X, Y Z)$ denotes the distance from point $X$ to the line $YZ$. Prove, that triangle $ABC$ is equilateral.
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) $p_1+2p_2$
1999 Tournament Of Towns, 4
A black unit equilateral triangle is drawn on the plane. How can we place nine tiles, each a unit equilateral triangle, on the plane so that they do not overlap, and each tile covers at least one interior point of the black triangle?
(Folklore)
2021 India National Olympiad, 5
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]
2005 May Olympiad, 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.
2005 JBMO Shortlist, 5
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
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?
2022 Iranian Geometry Olympiad, 5
a) Do there exist four equilateral triangles in the plane such that each two have
exactly one vertex in common, and every point in the plane lies on the boundary of at most two
of them?
b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them?
(Note that in both parts, there is no assumption on the intersection of interior of polygons.)
[i]Proposed by Hesam Rajabzadeh[/i]
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?
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.
2020 Caucasus Mathematical Olympiad, 7
A regular triangle $ABC$ is given. Points $K$ and $N$ lie in the segment $AB$, a point $L$ lies in the segment $AC$, and a point $M$ lies in the segment $BC$ so that $CL=AK$, $CM=BN$, $ML=KN$. Prove that $KL \parallel MN$.
1986 IMO, 2
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.