We study different types of electoral systems in the world, their features, advantages and problems. Legal election technologies for manipulating election results and ways to detect and eliminate the influence of such technologies are also considered. In addition, some results related to so-called "fair elections" are considered.

The paper considers a specific type of such financial instrument as an option, namely an exotic barrier call option of the European type. Exotic options are gaining popularity among ordinary investors due to the development of information and telecommunication technologies, thanks to which such specific financial instruments as options have become readily available. We investigate the hedging problem for such options with some restrictions on the payment function and the availability of dividend payment on a risky asset in the classical Black-Scholes model. An analogue of the Black-Scholes formula for the mentioned variant of the exotic barrier is proved. In the future, it is planned to generalize the obtained results for put options and for more general payment functions.

We study the theory of combinatorial games and find winning strategies for players. The algorithmic implementation of the winning strategies for the game TacTix is presented and the software implementation for this game in Python programming language is implemented. The program has a console interface and allows one to check the winning strategies in practice.

In this paper, a new method of detection of election fraud is proposed. This method is based on the calculation of the ratio of two standard normal random variables; estimation of parameters of obtained sample and comparison of these estimates with known theoretical values of parameters. Also in the paper, there is an example of the application of the described method.

We study the using the Monte Carlo method and its application. Below are several examples of software implementations of the Monte Carlo method for performing calculations that will allow us to determine the necessary information in cases where probability can be applied. Below is a software implementation of the examples in the C# programming language. The programs have a desktop interface and allow us to calculate such values as the number π and the time required to perform certain actions.