All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.

Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.

Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

The quadrilateral $ABCD$ is inscribed in a circle and has perpendicular diagonals. Points $K,L,M,Q$ are the points of intersection of the altitudes of the triangles $ABD, ACD, BCD, ABC$, respectively. Prove that the quadrilateral $KLMQ$ is equal to the quadrilateral $ABCD$. (Rozhkova Maria)

Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.

Let $D$ be a dodecahedron which can be inscribed in a sphere with radius $R$. Let $I$ be an icosahedron which can also be inscribed in a sphere of radius $R$. Which has the greater volume, and why? Note: A regular [i]polyhedron [/i] is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with $12$ faces which are regular pentagons and a regular icosahedron is a polyhedron with $20$ faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere. The illustration below shows a dodecahdron and an icosahedron, not necessarily to scale. [img]https://cdn.artofproblemsolving.com/attachments/7/5/9873b42aacf04bb5daa0fe70d4da3bf0b7be38.png[/img]

A trapezoid and an isosceles triangle are inscribed in a circle. The larger base of the trapezoid is the diameter of the circle, and the sides of the triangle are parallel to the sides of the trapezoid. Show that the trapezoid and the triangle have equal areas.

Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)

Let $ABCD$ be a trapezoid inscribed in a unit circle with diameter $AB$. If $DC = 4AD$, compute $AD$.

Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.

Find the side of the smallest regular triangle that can be inscribed in a right triangle with an acute angle of $30^o$ and a hypotenuse of $2$. (All vertices of the required regular triangle must be located on different sides of this right triangle.)

Projections of two points to the sidelines of a quadrilateral lie on two concentric circles (projections of each point form a cyclic quadrilateral and the radii of circles are different). Prove that this quadrilateral is a parallelogram.

Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle. (V. V . Proizvolov , Moscow)

A parallelogram is inscribed in a quadrilateral, two opposite vertices of which are the midpoints of the opposite sides of the quadrilateral. Determine the area of ​​such a parallelogram if the area of ​​the quadrilateral is equal to $S_o$.

What is a centrally symmetric polygon of greatest area one can inscribe in a given triangle?

A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.

A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.

Around the convex quadrilateral $ABCD$, three rectangles are circumscribed . It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square? (A rectangle is circumscribed around the quadrilateral $ABCD$ if there is one vertex $ABCD$ on each side of the rectangle).

It is given a triangle $\triangle ABC$. Determine the locus of center of rectangle inscribed in triangle $ABC$ such that one side of rectangle lies on side $AB$.

Points $ A_1 $, $ B_1 $, $ C_1 $ divide respectively the sides $ BC $, $ CA $, $ AB $ of the triangle $ ABC $ in the ratios $ k_1 $, $ k_2 $, $ k_3 $. Calculate the ratio of the areas of triangles $ A_1B_1C_1 $ and $ ABC $.

Given a convex quadrilateral whose opposite sides are not parallel, and giving an internal point $P$. Find a parallelogram whose vertices are on the side lines of the rectangle and whose center is $P$. Give a method by which we can construct it (provided there is one). [img]https://1.bp.blogspot.com/-t4aCJza0LxI/X9j1qbSQE4I/AAAAAAAAMz4/V9pr7Cd22G4F320nyRLZMRnz18hMw9NHQCLcBGAsYHQ/s0/2012%2BDurer%2BC3.png[/img]

We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .

In the triangle $ ABC $ on the sides $ BC $, $ CA $, $ AB $, the points $ D $, $ E $, $ F $ are chosen respectively in such a way that $$ BD \colon DC = CE \colon EA = AF \colon FB = k,$$ where $k$ is a given positive number. Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle $ DEF $

Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.

A rectangle, whose one sidelength is twice the other side, is inscribed inside a triangles with sides $3$ cm, $4$ cm and $5$ cm, such that the long sides lies entirely on the long side of the triangle. The other two remaining vertices of the rectangle lie respectively on the other two sides of the triangle. Find the lengths of the sides of this rectangle.