Olympiad problems

1963 German National Olympiad, 4

Consider a rectangle with sides $2a$ and $2b$, where $a > b$. There should be four congruent right triangles (one triangle at each vertex of this rectangle , whose legs are on the sides of the rectangle lie) must be cut off so that the remaining figure forms an octagon with sides of equal length. The side of the octagon is to be expressed in terms of a and $b$ and constructed from $a$ and $b$. Besides that it must be stated under which conditions the problem can be solved.

rectangle octagon geometry number theory

Durer Math Competition CD Finals - geometry, 2008.C3

We divided a regular octagon into parallelograms. Prove that there are at least $2$ rectangles between the parallelograms.

octagon geometry parallelogram

Ukrainian From Tasks to Tasks - geometry, 2011.14

The lengths of the four sides of an cyclic octagon are $4$ cm, the lengths of the other four sides are $6$ cm. Find the area of the octagon.

cyclic quadrilateral octagon geometry cyclic quadrilaterals