The truth of the Olympic content inequalities about one method of approval

The paper discusses certain categories of inequalities and one non-standard method of solving/proving them, which implies that if the inequality fa+fb+fc+…≥1, is to be proved, then we will try to choose a number k such that fair let there be inequalities: faakak+bk+ck+…; fbbkak+bk+ck+…; fcckak+bk+ck+… and then we get the desired result as a result of adding these true inequalities. It should be noted that the first two of the problems discussed in the presented paper are quite well-known inequalities (Nesbitt's inequality and the problem of the 2001 International Mathematical Olympiad), and the third one we created specifically to offer more practical exercises to the reader so that he can get to know the presented method better.