By Pak I.

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C) [1] Same for inscribed rhombi with an edge parallel to a given line. d) [1] Find examples of polygons with parallel edges which do have inscribed rhombi as in parts a) and b). Prove that whenever a polygon has two such rhombi, it has inﬁnitely many of them. e) [1-] Find a convex polygon X ⊂ R2 with non-parallel edges and a vertex v, such that X has exactly two inscribed rhombi with vertices at v. 3. a) [1] Let X ⊂ R2 be a simple polygon. Prove that there exist a cyclic quadrilateral (inscribed into a circle) which is inscribed into X, and whose vertices divide closed curve X into four arcs of equal length.

Compute the number of quadrilaterals similar to Q, with all vertices on lines ℓi . 18. e. coplanar) rhombus. b) [1-] Prove that every ﬂat rhombus inscribed into a sphere is a square. Use part a) to conclude that every simple spherical polygon has an inscribed square. c) [2-] Use part b) and a limit argument to give another proof that every simple polygon in the plane has an inscribed square. 19. An equihedral octahedron Q is deﬁned as convex hull of three orthogonal intervals (diagonals of Q) intersecting at midpoints.

This follows from the fact that (0, 1) and (1, 0) lie in different connected components of [0, 1]2 A. Indeed, if one climber is going up the mountain and the other is going down, there is a time t when they are at the same level. Second proof. e. all peaks and valleys of the mountain are at different levels. Define the set A as in the first proof. Let us show that A is a continuous piecewise linear curve from (0, 0) to (1, 1). 8). We conclude that A is a continuous curve with (0, 0) and (1, 1) its only possible endpoints.