This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 296

1991 Swedish Mathematical Competition, 6
Tags: geometry , area , geometric inequalities , equilateral
Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

2002 Junior Balkan Team Selection Tests - Romania, 3
Tags: combinatorics , combinatorial geometry , equilateral
A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines.

2023 Regional Olympiad of Mexico West, 5
Tags: geometry , rhombus , equilateral , parallelogram
We have a rhombus $ABCD$ with $\angle BAD=60^\circ$. We take points $F,H,G$ on the sides $AD,DC$ and the diagonal $AC$, respectively, such that $DFGH$ is a parallelogram. Prove that $BFH$ is equilateral.

2007 Singapore Senior Math Olympiad, 3
Tags: geometry , equilateral triangle , equilateral , circumcircle
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 $PA^2 \cdot QB = QA^2 \cdot PB$.

Oliforum Contest V 2017, 2
Tags: square , equilateral , tiling , combinatorics , combinatorial geometry
Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$. (Emanuele Tron)

2012 Czech-Polish-Slovak Junior Match, 3
Tags: projection , circumcircle , equilateral , fixed , area of a triangle , geometry
Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that (a) the triangle $KLM$ is equilateral, (b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$

Estonia Open Junior - geometry, 2019.2.1
Tags: geometry , pentagon , equilateral , angle
A pentagon can be divided into equilateral triangles. Find all the possibilities that the sizes of the angles of this pentagon can be.

1967 All Soviet Union Mathematical Olympiad, 084
Tags: geometry , altitude , median , equilateral
a) The maximal height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$. Prove that the angle $ABC$ isn't greater than $60$ degrees. b) The height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$ and bisectrix $|CD|$. Prove that the angle $ABC$ is 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 $PO$.

2002 All-Russian Olympiad Regional Round, 8.4
Tags: geometry , equilateral
Given a triangle $ABC$ with pairwise distinct sides. on his on the sides, regular triangles $ABC_1$, $BCA_1$, $CAB_1$. are constructed externally. Prove that triangle $A_1B_1C_1$ cannot be regular.

2018 Brazil EGMO TST, 3
Tags: geometry , equilateral , concurrency
An equilateral triangle $ABC$ is inscribed in a circle $\Omega$ and has incircle $\omega$. Points $P$ and $Q$ are in segments $AC$ and $AB$, respectively, such that $PQ$ is tangent to $\omega$. The circle $\Omega_B$ has center $P$ and radius $PB$ and the circle $\Omega_C$ is defined similarly. Prove that $\Omega$, $\Omega_B$ and $\Omega_C$ have a common point.

2017 Hanoi Open Mathematics Competitions, 11
Tags: geometry , inequalities , geometric inequality , equilateral , angle
Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle. Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?

1961 Czech and Slovak Olympiad III A, 4
Tags: equilateral , triangle , computation , triangle area , max , min , geometry
Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.

Durer Math Competition CD Finals - geometry, 2014.C2
Tags: geometry , equilateral , fixed
Let $P$ be an arbitrary interior point of the equilateral triangle $ABC$. From $P$ draw parallel to the sides: $A'_1A_1 \parallel AB$, $B' _1B_1 \parallel BC$ and $C'_1C_1 \parallel CA$. Prove that the sum of legths $| AC_1 | + | BA_1 | + | CB_1 |$ is independent of the choice of point $P$. [img]https://cdn.artofproblemsolving.com/attachments/5/a/15b06706c09e2458fb5938807b9f3833ffb62e.png[/img]

2012 Austria Beginners' Competition, 4
Tags: equilateral , geometry , trisect
A segment $AB$ is given. We erect the equilateral triangles $ABC$ and $ADB$ above and below $AB$. We denote the midpoints of $AC$ and $BC$ by $E$ and $F$ respectively. Prove that the straight lines $DE$ and $DF$ divide the segment $AB$ into three parts of equal length .

2017 Hanoi Open Mathematics Competitions, 9
Tags: geometry , equilateral
Prove that the equilateral triangle of area $1$ can be covered by five arbitrary equilateral triangles having the total area $2$.

2009 Moldova National Olympiad, 9.3
Tags: equilateral , parallel , geometry
Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.

Estonia Open Junior - geometry, 2003.2.2
Tags: geometry , equilateral
The shape of a dog kennel from above is an equilateral triangle with side length $1$ m and its corners in points $A, B$ and $C$, as shown in the picture. The chain of the dog is of length $6$ m and its end is fixed to the corner in point $A$. The dog himself is in point $K$ in a way that the chain is tight and points $K, A$ and $B$ are on the same straight line. The dog starts to move clockwise around the kennel, holding the chain tight all the time. How long is the walk of the dog until the moment when the chain is tied round the kennel at full? [img]https://cdn.artofproblemsolving.com/attachments/9/5/616f8adfe66e2eb60f1a6c3f26e652c45f3e27.png[/img]

2006 Oral Moscow Geometry Olympiad, 6
Tags: geometry , equilateral , circumcenter , centroid
In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it. (A. Zaslavsky)

2004 Junior Balkan Team Selection Tests - Romania, 2
Tags: midpoint , geometry , equilateral , cyclic quadrilateral , centroid
Let $M,N, P$ be the midpoints of the sides $BC,CA,AB$ of the triangle $ABC$, respectively, and let $G$ be the centroid of the triangle. Prove that if $BMGP$ is cyclic and $2BN = \sqrt3 AB$ , then triangle $ABC$ is equilateral.

Novosibirsk Oral Geo Oly IX, 2016.6
Tags: geometry , equal segments , equilateral
An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

1998 Belarus Team Selection Test, 3
Tags: geometry , parallelogram , hexagon , area , equilateral
Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

1981 All Soviet Union Mathematical Olympiad, 309
Tags: equilateral , geometry , vector
Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral

2015 Oral Moscow Geometry Olympiad, 5
Tags: geometric inequality , rhombus , equilateral , geometry
On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.

2007 Singapore Junior Math Olympiad, 2
Tags: geometry , equilateral , parallelogram , equilateral triangle
Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.