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#### Convex polygons , Pigeonhole principle (finite number of poits, lines etc.)

What is the minimum number of points necessary to mark inside a convex $n$-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?

#### Continuity considerations , Convex polygons , Intermediate value theorem. Connectedness.

A convex figure and point A inside it are given. Prove that there is a chord $($that is, a segment joining two boundary points of a convex figure$)$ passing through point A and dividing it in half at point A.

#### Convex polygons , Pentagons , Pigeonhole principle (finite number of poits, lines etc.) , Plane dissected by lines

Two points are placed inside a convex pentagon. Prove that it is always possible to choose a quadrilateral that shares four of the five vertices on the pentagon, such that both of the points lie inside the quadrilateral.

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