Problems

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There are 5 points inside an equilateral triangle with side of length 1. Prove that the distance between some two of them is less than 0.5.

There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.

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?

A plane contains \(n\) straight lines, of which no two are parallel. Prove that some of the angles will be smaller than \(180^\circ/n\).

Cut an arbitrary triangle into 3 parts and out of these pieces construct a rectangle.