Olympiad problems

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 $PO$.

1996 Tournament Of Towns, (505) 2

Tags: trapezoid , Equilateral , tiles , geometry , combinatorics , combinatorial geometry

For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$? (NB Vassiliev)

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 $AM, BM, CM$.

2006 Portugal MO, 2

Tags: geometry , Equilateral , perpendicular

In the equilateral triangle $[ABC], D$ is the midpoint of $[AC], E$ and the orthogonal projection of $D$ over $[CB]$ and $F$ is the midpoint of $[DE]$. Prove that $[FB]$ and $[AE]$ are perpendicular. [img]https://1.bp.blogspot.com/-TjSyQotGIOM/X4XMolaXHvI/AAAAAAAAMng/cVsHfl-lrXAFE5LMdosE6vqK1Tf-8WOQgCLcBGAsYHQ/s0/2006%2Bportugal%2Bp2.png[/img]

1972 Spain Mathematical Olympiad, 5

Tags: geometry , construction , Equilateral

Given two parallel lines $r$ and $r'$ and a point $P$ on the plane that contains them and that is not on them, determine an equilateral triangle whose vertex is point $P$ , and the other two, one on each of the two lines. [img]https://cdn.artofproblemsolving.com/attachments/9/3/1d475eb3e9a8a48f4a85a2a311e1bda978e740.png[/img]

Kyiv City MO 1984-93 - geometry, 1985.9.5

Tags: geometry , parallelogram , Equilateral

Outside the parallelogram $ABCD$ on its sides $AB$ and $BC$ are constructed equilateral triangles $ABK$, and $BCM$. Prove that the triangle $KMD$ is equilateral.

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.

1985 Swedish Mathematical Competition, 3

Tags: geometry , isosceles , Equilateral

Points $A,B,C$ with $AB = BC$ are given on a circle with radius $r$, and $D$ is a point inside the circle such that the triangle $BCD$ is equilateral. The line $AD$ meets the circle again at $E$. Show that $DE = r$.

1999 Swedish Mathematical Competition, 4

Tags: geometry , Equilateral

An equilateral triangle of side $x$ has its vertices on the sides of a square side $1$. What are the possible values of $x$?

2016 Novosibirsk Oral Olympiad in Geometry, 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 $AM, BM, CM$.

2005 Sharygin Geometry Olympiad, 7

Tags: Equilateral , Equilateral Triangle , concentric circles , geometry , angles

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?

2016 BMT Spring, 3

Tags: geometry , Equilateral , square

Consider an equilateral triangle and square, both with area $1$. What is the product of their perimeters?