Found problems: 287
Cono Sur Shortlist - geometry, 1993.8
In a triangle $ABC$, let $D$, $E$ and $F$ be the touchpoints of the inscribed circle and the sides $AB$, $BC$ and $CA$. Show that the triangles $DEF$ and $ABC$ are similar if and only if $ABC$ is equilateral.
VII Soros Olympiad 2000 - 01, 8.4
Paint the maximum number of vertices of the cube red so that you cannot select three of the red vertices that form an equilateral triangle.
2020 Malaysia IMONST 2, 2
Prove that for any integer $n\ge 6$ we can divide an equilateral triangle completely into $n$ smaller equilateral triangles.
1995 Bulgaria National Olympiad, 4
Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.
1989 Tournament Of Towns, (229) 3
The plane is cut up into equilateral triangles by three families of parallel lines.
Is it possible to find $4$ vertices of these triangles which form a square?
Kyiv City MO Seniors Round2 2010+ geometry, 2015.11.2
The line passing through the center of the equilateral triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ {{C} _ {1}} $, $ {{A} _ {1}} $ and $ {{B} _ {1}} $, respectively. Let $ {{A} _ {2}} $ be a point that is symmetric $ {{A} _ {1}} $ with respect to the midpoint of $ BC $; the points $ {{B} _ {2}} $ and $ {{C} _ {2}} $ are defined similarly. Prove that the points $ {{A} _ {2}} $, $ {{B} _ {2}} $ and $ {{C} _ {2}} $ lie on the same line tangent to the inscribed circle of the triangle $ ABC $.
(Serdyuk Nazar)
May Olympiad L1 - geometry, 1998.4
$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle.
[img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]
2011 Tournament of Towns, 3
(a) Does there exist an in nite triangular beam such that two of its cross-sections are similar but not congruent triangles?
(b) Does there exist an in nite triangular beam such that two of its cross-sections are equilateral triangles of sides $1$ and $2$ respectively?
Ukrainian TYM Qualifying - geometry, VI.2
Let $A_1,B_1,C_1$ be the midpoints of the sides of the $BC,AC, AB$ of an equilateral triangle $ABC$. Around the triangle $A_1B_1C_1$ is a circle $\gamma$, to which the tangents $B_2C_2$, $A_2C_2$, $A_2B_2$ are drawn, respectively, parallel to the sides $BC, AC, AB$. These tangents have no points in common with the interior of triangle $ABC$. Find out the mutual location of the points of intersection of the lines $AA_2$ and $BB_2$, $AA_2$ and $CC_2$, $BB_2$ and $CC_2$ and the circumscribed circle $\gamma$. Try to consider the case of arbitrary points $A_1,B_1,C_1$ located on the sides of the triangle $ABC$.
Kyiv City MO 1984-93 - geometry, 1993.9.3
The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.
1999 Kazakhstan National Olympiad, 3
The circle inscribed in the triangle $ ABC $ , with center $O$, touches the sides $ AB $ and $ BC $ at the points $ C_1 $ and $ A_1 $, respectively. The lines $ CO $ and $ AO $ intersect the line $ C_1A_1 $ at the points $ K $ and $ L $. $ M $ is the midpoint of $ AC $ and $ \angle ABC = 60^\circ $. Prove that $ KLM $ is a regular triangle.
2004 All-Russian Olympiad Regional Round, 8.3
In an acute triangle, the distance from the midpoint of any side to the opposite vertex is equal to the sum of the distances from it to sides of the triangle. Prove that this triangle is equilateral.
Kyiv City MO Juniors Round2 2010+ geometry, 2019.9.3
The equilateral triangle $ABC$ is inscribed in the circle $w$. Points $F$ and $E$ on the sides $AB$ and $AC$, respectively, are chosen such that $\angle ABE+ \angle ACF = 60^o$. The circumscribed circle of $\vartriangle AFE$ intersects the circle $w$ at the point $D$ for the second time. The rays $DE$ and $DF$ intersect the line $BC$ at the points $X$ and $Y$, respectively. Prove that the center of the inscribed circle of $\vartriangle DXY$ does not depend on the choice of points $F$ and $E$.
(Hilko Danilo)
1997 Abels Math Contest (Norwegian MO), 2a
Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.
2014 Oral Moscow Geometry Olympiad, 5
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 $A'B'C'$.
1997 Singapore MO Open, 1
$\vartriangle ABC$ is an equilateral triangle. $L, M$ and $N$ are points on $BC, CA$ and $AB$ respectively. Prove that $MA \cdot AN + NB \cdot BL + LC \cdot CM < BC^2$.
1980 All Soviet Union Mathematical Olympiad, 298
Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ be the midpoint of the $[AP]$ segment. Find the angles of triangle $DEC$ .
Champions Tournament Seniors - geometry, 2016.3
Let $t$ be a line passing through the vertex $A$ of the equilateral $ABC$, parallel to the side $BC$. On the side $AC$ arbitrarily mark the point $D$. Bisector of the angle $ABD$ intersects the line $t$at the point $E$. Prove that $BD=CD+AE$.
2006 Sharygin Geometry Olympiad, 8.1
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
2022 New Zealand MO, 1
$ABCD$ is a rectangle with side lengths $AB = CD = 1$ and $BC = DA = 2$. Let $ M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\angle PCB$.
2019 Saudi Arabia Pre-TST + Training Tests, 2.3
Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that
i) $AX = BY = CZ$
ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order.
Prove that $XY Z$ is an equilateral triangle.
1961 All Russian Mathematical Olympiad, 006
a) Points $A$ and $B$ move uniformly and with equal angle speed along the circumferences with $O_a$ and $O_b$ centres (both clockwise). Prove that a vertex $C$ of the equilateral triangle $ABC$ also moves along a certain circumference uniformly.
b) The distance from the point $P$ to the vertices of the equilateral triangle $ABC$ equal $|AP|=2, |BP|=3$. Find the maximal value of $CP$.
Kyiv City MO Juniors Round2 2010+ geometry, 2021.7.4
The sides of the triangle $ABC$ are extended in both directions and on these extensions $6$ equal segments $AA_1 , AA_2, BB_1,BB_2, CC_1, CC_2$ are drawn (fig.). It turned out that all $6$ points $A_1,A_2,B_1,B_2,C_1, C_2$ lie on the same circle, is $\vartriangle ABC$ necessarily equilateral?
(Bogdan Rublev)
[img]https://cdn.artofproblemsolving.com/attachments/0/3/a499f6e6d978ce63d2ab40460dc73b62882863.png[/img]
Estonia Open Senior - geometry, 2014.1.4
In a plane there is a triangle $ABC$. Line $AC$ is tangent to circle $c_A$ at point $C$ and circle $c_A$ passes through point $B$. Line $BC$ is tangent to circle $c_B$ at point $C$ and circle $c_B$ passes through point $A$. The second intersection point $S$ of circles $c_A$ and $c_B$ coincides with the incenter of triangle $ABC$. Prove that the triangle $ABC$ is equilateral.
1997 All-Russian Olympiad Regional Round, 8.3
On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.