Modern mathematical ideas are quite difficult for even high school students to understand from both a mathematical and terminological point of view. The exception may be an algorithm Gale-Shapley (algorithm for creating persistent pairs), who won the Nobel Prize in 2012 in the brunch of economy. This algorithm is interesting in that it shows the modern application of mathematics to solving specific practical problems (admission to universities, selection of donors, and selection of applicants for positions). There are considered three problems, which that are solved using the Gale-Shapley algorithm such as the problem of marriage (marriage), the problem of choosing fruits, and the problem of admission in this article. The first problem is a classic example of the implementation of the algorithm Gale-Shapley’s steps on its example we will consider features of this algorithm, the differences between "male" and "female" distributions, and the evaluation optimality of these distributions. The second problem shows how the algorithm changes if the elements of one of the sets cannot influence the distribution. In the third problem, the example of real people - graduates school № 6 of Lebedyn (Sumy region, Ukraine), shows how the algorithm works Gale-Shapley in modern Ukrainian system of distribution of students in the universities. There are also stated methodical features the use of such tasks in extracurricular activities in mathematics and benefit from it in the article. Knowledge of the algorithm for creating stable pairs, which are consider in the article, will be useful to every mathematics teacher and students, because the Gale-Shapley algorithm demonstrates the application of mathematics to real processes in different brunch, deepening the understanding of the processes themselves, and the knowledge gained is modern and relevant.