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Given 50 segments on the line. Prove that one of the following statements is valid: 1. Some 8 segments have a common point. 2. Some 8 segments do not intersect each other.

Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons.

n people party at a restaurant, sitting at the vertices of an n-gon-shaped dinner table. Their orders have been mixed up -- in fact, none of them have received the correct entree. Show that the table may be rotated so that at least two people are sitting in front of the correct entree.

@Perhaps check it again I thought of that distinction as well, and it led me back to the OP, where I found that the word "interior" is not mentioned. It has been agreed that a line cannot always pass through the interior of polygons so described. It has been clarified that BMAD asks something different. Not all Denizens share English as their primary language, so discussions like this one are generally helpful if taken in that regard.

I claim it's not true, and I show it with a counterexample. In the xy-plane, let the following points be labelled as such: A = (0, 1) B = (1, 1) C = (-1, 0) D = (0, 0) E = (1, 0) F = (0, -1) Triangle ABE shares point A with triangle ACD. Triangle ACD shares point D with triangle EDF. Triangle ABE shares point E with triangle EDF. No line can pass through all the interiors of the three triangles.

Given five circles where any group of 4 have a common point. Does there exist a common point for all 5?

I stated "No line can pass through all the interiors of the the three triangles." BMD, you stated "Why cant [sic] a line pass through [triangle] ABE and the common point of [triangles] ACD and EDF?" In my attempt to work on the problem, I misunderstood "crossing a polygon" from the original post and changed the subject. But, with you asking me that question (in the quotes just above), you changed the subject. I never addressed/denied that situation. It's not a line passing through the interiors of three triangles anyway.