This paper presents an example of solutions of elementary mathematical problems designed for high school students. The focus was on simple tasks of algebraic and geometrical kind, whose solving may lead to systems of nonlinear equations, in which variables are bound by special conditions. Symmetric polynomials are not a part of curriculum of high school mathematics, but students can already encounter Viete's formulas at this level in the connection with the study of quadratic equations. From Vieta's formulas there is only a small step to the definition of symmetric polynomials and the derivation of Newton's formulas. A wide variety of examples can be solved by Newton's formulas together with Vieta's formulas very easily and elegantly. New topics can bring the enrichment for high school mathematics mostly at schools with an extended teaching mathematics.