Olympiad problems

2011 Sharygin Geometry Olympiad, 13

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).

tetrahedron inscribed triangle geometry 3d geometry centroid locus solid geometry

2009 Bosnia And Herzegovina - Regional Olympiad, 1

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

inscribed triangle geometry

2006 Sharygin Geometry Olympiad, 8.1

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

perimeter construction inscribed triangle equilateral triangle geometry equilateral maximum equilateral triangles geometric constructions

1976 Euclid, 9

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 $\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } \text{none of these}$

inscribed triangle geometry law of cosines

2011 Oral Moscow Geometry Olympiad, 6

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.

geometric inequality inscribed triangle geometry inscribed side geometric inequalities