Two identical gears have 32 teeth. They were combined and 6 pairs of teeth were simultaneously removed. Prove that one gear can be rotated relative to the other so that in the gaps in one gear where teeth were removed the second gear will have whole teeth.
4 points \(a, b, c, d\) lie on the segment \([0, 1]\) of the number line. Prove that there will be a point \(x\), lying in the segment \([0, 1]\), that satisfies \[\frac{1}{ | x-a |}+\frac{1}{ | x-b |}+\frac{1}{ | x-c |}+\frac{1}{ | x-d |} < 40.\]
Prove that any convex polygon contains not more than \(35\) vertices with an angle of less than \(170^\circ\).
A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than \(720^{\circ}\).
Prove that it is not possible to completely cover an equilateral triangle with two smaller equilateral triangles.
On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.
12 straight lines passing through the origin are drawn on a plane. Prove that it is possible to choose two of these lines such that the angle between them is less than 17 degrees.
A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?
There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.
Some open sectors – that is sectors of circles with infinite radii – completely cover a plane. Prove that the sum of the angles of these sectors is no less than \(360^\circ\).