## 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:

**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.
**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*
**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*.
**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 ~ Force^{n}

**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.