Artyom Sokirko's unsolved problems

Here is the list of the unsolved problems that I am interested in. I started to collect these problems over 20 years ago. Actually, it is not quite true: I do not collect problems, but they come and stay against my will. Someone told me that the best way to handle such intruders is to put them on paper - but it is does not help, because they are too beautiful:

1. Four-angles on the chessboard (1977). Prove that a four-angle of any size and shape can be placed on an unlimited chessboard in such way that all four of its apexes fall within cells of the same color.
2. A car on a conical surface (1982). A car is moving in horizontal circle on the external surface of standing up-right cone. At what speed v will the car
   a) start to skid down
   b) go up into the air.
   The angle between the cone's surface and the horizontal plane is alpha, the friction coefficient is um, the radius of cycle is R.
   For the car on the external surface of a cone:
   c) when will it start to skid down
   d) skid up?
   As you can see, this is a purely cinematic problem
3. Sphere under pressure (1994). What is the maximum of external pressure that a thin sphere can withstand? The sphere's radius is R, the shell's thickness is h, (h << R), the elasticity module is mu.
4. Fractal cube elasticity (1998). Fractal with dimension m the external form of cube is made of a regular material as limit transition. Find the primary [power] coefficient n in quasi-Huke relation:

   Deformation ~ Forceⁿ

5. Light speed is the function of space coordinates - a new model of the Universe, please see details.

If you have any comments about this list, please drop me a line.