Olympiad problems

2011 Oral Moscow Geometry Olympiad, 6

Tags: side , inscribed triangle , inscribed , geometry , geometric inequality

One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.

2011 Sharygin Geometry Olympiad, 13

Tags: tetrahedron , locus , centroid , inscribed triangle , 3d geometry , geometry

a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex). b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).

2006 Sharygin Geometry Olympiad, 8.1

Tags: construction , perimeter , equilateral , equilateral triangle , inscribed triangle , maximum , geometry

Inscribe the equilateral triangle of the largest perimeter in a given semicircle.

2009 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: geometry , inscribed triangle

In triangle $ABC$ holds $\angle ACB = 90^{\circ}$, $\angle BAC = 30^{\circ}$ and $BC=1$. In triangle $ABC$ is inscribed equilateral triangle (every side of a triangle $ABC$ contains one vertex of inscribed triangle). Find the least possible value of side of inscribed equilateral triangle

1976 Euclid, 9

Tags: inscribed triangle , geometry

Source: 1976 Euclid Part A Problem 9 ----- A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. The measure of the angle subtended at the centre of the circle by the shortest side is $\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } \text{none of these}$