Olympiad problems

1998 Chile National Olympiad, 6

Given an equilateral triangle, cut it into four polygonal shapes so that, reassembled appropriately, these figures form a square.

2002 Tuymaada Olympiad, 2

Tags: geometry , ratio , equilateral , equilateral triangle

Points on the sides $ BC $, $ CA $ and $ AB $ of the triangle $ ABC $ are respectively $ A_1 $, $ B_1 $ and $ C_1 $ such that $ AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1 $. Prove that if triangle $ A_1B_1C_1 $ is equilateral, then triangle $ ABC $ is also equilateral.

1999 Abels Math Contest (Norwegian MO), 3

Tags: isosceles , equilateral , triangle area , area

An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$. Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$, and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$. The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$, respectively. (a) Prove that triangle $AFG$ is equilateral. (b) Find the ratio between the areas of triangles $AFE$ and $ABC$.

Kyiv City MO 1984-93 - geometry, 1989.8.5

Tags: geometry , right triangle , equilateral , inscribed , construction

The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.

2012 Tournament of Towns, 4

Tags: geometry , incenter , equilateral

Given a triangle $ABC$. Suppose I is its incentre, and $X, Y, Z$ are the incentres of triangles $AIB, BIC$ and $AIC$ respectively. The incentre of triangle $XYZ$ coincides with $I$. Is it necessarily true that triangle $ABC$ is regular?

1999 Bundeswettbewerb Mathematik, 3

Tags: isosceles , equilateral , geometry

In the plane are given a segment $AC$ and a point $B$ on the segment. Let us draw the positively oriented isosceles triangles $ABS_1, BCS_2$, and $CAS_3$ with the angles at $S_1,S_2,S_3$ equal to $120^o$. Prove that the triangle $S_1S_2S_3$ is equilateral.

1998 May Olympiad, 4

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

$ABCD$ 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]

1997 Tournament Of Towns, (556) 6

Tags: geometry , combinatorics , combinatorial geometry , equilateral

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into $10$ equal segments and the triangle into $100$ congruent triangles. Each of these $100$ triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. What is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (R Zhenodarov)

1990 Tournament Of Towns, (277) 2

Tags: geometry , equilateral

A point $M$ is chosen on the arc $AC$ of the circumcircle of the equilateral triangle $ABC$. $P$ is the midpoint of this arc, $N$ is the midpoint of the chord $BM$ and $K$ is the foot of the perpendicular drawn from $P$ to $MC$. Prove that the triangle $ANK$ is equilateral. (I Nagel, Yevpatoria)

2005 Peru MO (ONEM), 3

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

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 $\angle MBN$.

2006 Oral Moscow Geometry Olympiad, 6

Tags: equilateral , circumcenter , centroid , geometry

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)

1973 Chisinau City MO, 68

Tags: geometry , equilateral , similar

Inside the triangle $ABC$, point $O$ was chosen so that the triangles $AOB, BOC, COA$ turned out to be similar. Prove that triangle $ABC$ is equilateral.

Ukrainian TYM Qualifying - geometry, V.8

Tags: geometry , trigonometry , geometric inequality , equilateral

Let $X$ be a point inside an equilateral triangle $ABC$ such that $BX+CX <3 AX$. Prove that $$3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5$$

2007 Sharygin Geometry Olympiad, 1

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

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.

Oliforum Contest V 2017, 2

Tags: tiling , combinatorics , combinatorial geometry , square , equilateral

Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$. (Emanuele Tron)

2002 District Olympiad, 3

Tags: geometry , midpoint , equilateral , centroid , distance

Consider the equilateral triangle $ABC$ with center of gravity $G$. Let $M$ be a point, inside the triangle and $O$ be the midpoint of the segment $MG$. Three segments go through $M$, each parallel to one side of the triangle and with the ends on the other two sides of the given triangle. a) Show that $O$ is at equal distances from the midpoints of the three segments considered. b) Show that the midpoints of the three segments are the vertices of an equilateral triangle.

2012 Tournament of Towns, 7

Tags: incenter , angle , equilateral , altitude

Let $AH$ be an altitude of an equilateral triangle $ABC$. Let $I$ be the incentre of triangle $ABH$, and let $L, K$ and $J$ be the incentres of triangles $ABI,BCI$ and $CAI$ respectively. Determine $\angle KJL$.

Novosibirsk Oral Geo Oly VII, 2021.6

Tags: geometry , equal segments , equilateral

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

1955 Moscow Mathematical Olympiad, 289

Tags: geometry , locus , equilateral , equal segments

Consider an equilateral triangle $\vartriangle ABC$ and points $D$ and $E$ on the sides $AB$ and $BC$csuch that $AE = CD$. Find the locus of intersection points of $AE$ with $CD$ as points $D$ and $E$ vary.

1987 Tournament Of Towns, (148) 5

Tags: geometry , locus , perpendicular , equilateral

Perpendiculars are drawn from an interior point $M$ of the equilateral triangle $ABC$ to its sides , intersecting them at points $D, E$ and $F$ . Find the locus of all points $M$ such that $DEF$ is a right triangle . (J . Tabov , Sofia)

1994 Tournament Of Towns, (426) 3

Tags: geometry , equilateral

Two-mutually perpendicular lines $\ell$ and $m$ intersect each other at a point of the circumference of a circle, dividing it into three arcs. A point $M_i$ ($i = 1$,$2$,$3$) is taken on each arc so that the tangent line to the circumference at the point $M_i$ intersects $\ell$ and $m$ in two points at the same distance from $M_i$ (that is $M_i$ is the midpoint of the segment between them). Prove that the triangle $M_1M_2M_3$ is equilateral. (Przhevalsky)

2005 Switzerland - Final Round, 1

Tags: geometry , equilateral , centroid , concyclic , cyclic quadrilateral , median

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

2020 Puerto Rico Team Selection Test, 1

Tags: combinatorial geometry , combinatorics , hexagon , equilateral

We have $10,000$ identical equilateral triangles. Consider the largest regular hexagon that can be formed with these triangles without overlapping. How many triangles will not be used?

2019 District Olympiad, 4

Tags: geometry , equilateral , right triangle , isosceles

Consider the isosceles right triangle$ ABC, \angle A = 90^o$, and point $D \in (AB)$ such that $AD = \frac13 AB$. In the half-plane determined by the line $AB$ and point $C$ , consider a point $E$ such that $\angle BDE = 60^o$ and $\angle DBE = 75^o$. Lines $BC$ and $DE$ intersect at point $G$, and the line passing through point $G$ parallel to the line $AC$ intersects the line $BE$ at point $H$. Prove that the triangle $CEH$ is equilateral.

2018 Malaysia National Olympiad, A1

Tags: geometry , hexagon , equilateral

Hassan has a piece of paper in the shape of a hexagon. The interior angles are all $120^o$, and the side lengths are $1$, $2$, $3$, $4$, $5$, $6$, although not in that order. Initially, the paper is in the shape of an equilateral triangle, then Hassan has cut off three smaller equilateral triangle shapes, one at each corner of the paper. What is the minimum possible side length of the original triangle?