## Sample problems

Below is a set of problems to demonstrate the level of the contest. Most problems from the list below have
appeared before and are presented here only as an example, the actual olympiad will consist of fully original problems.
The final set of problems will appear in the personal team page exactly at the start of the contest.

#### Two balls

*Landau's theoretical exam, classical mechanics*
Two balls (with masses *m* and *M*) are positioned along one line which is perpendicular to the massive wall.
The right ball moves to the left with velocity *v* (in the direction of the wall). Find the sum
*N*_{1} + *N*_{2}, where *N*_{1} is the total number of collision between the
left ball and the wall and *N*_{2} is the total number of collisions between the balls.

#### Metallic cavity

*Textbook problem*
Find the force acting on a positive charge *e* located at an arbitrary point in a cubic cavity (with edge *b*)
of a metallic medium.

#### Free falling charge

*The problem is recent and still scientic.*
A charge *e* with mass *m* is in a free fall in the gravitational field of Schwarzschild black hole.
For a static observer it moves with an acceleration. Would it be radiating according to the observer?

#### Hydrogen atom

*Textbook problem*
Let us consider **H** atom in a weak uniform magnetic field. Assume that an electron is in the ground state and
that the motion of a proton is adiabatically slow with respect to the electron. Find an effective Hamiltonian
describing the motion of the proton under these conditions and show that it behaves as a particle of zero charge.

#### Droplet

*Olympiad in memory of Landau*
In the closed bulb there is a vapor with density *n*_{1} (no other gases). Find the way how a droplet
hovering in the vapor grows with time. Suppose that near the surface of the droplet the density of vapor is
*n*_{0} (*n*_{0} < *n*_{1} and suppose *n*_{0} to be
the equilibrium density).

#### Entropic force

*Landau's theoretical exam, statistical mechanics*
There is a long polymer chain with *N* links each of length *l* at a temperature *T*. One end
of the chain is xed. Find the force required to keep the other end at the distance *L*. Assume that the
temperature is very large. *Bonus:* how large should it be?

*Hint:* read the title of the problem, carefully.