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For the differential system , , , where , , , we get necessary and sufficient conditions that this system does not have solutions satisfying the condition for . Note one of our results obtained for this system with constant coefficients and delays ( , where and ). The inequality is necessary and sufficient for nonexistence of solutions satisfying this condition.

The article discusses what the goals of mathematical competitions should be based on the objectives of mathematical education. It cites the mathematical olympiads which originated in Romania in the 1860s where the students' level of knowledge and their ability to think creatively when solving mathematical problems were tested through a game. It points to the Internet Mathematical Olympiad where people worldwide can compete with each other first held in 2006 for Israeli students.

In the classical theorems about lower and upper vector functions for systems of linear differential equations very heavy restrictions on the signs of coefficients are assumed. These restrictions in many cases become necessary if we wish to compare all the components of a solution vector. The formulas of the integral representation of the general solution explain that these theorems claim actually the positivity of all elements of the Green's matrix. In this paper we define a principle of partial monotonicity (comparison of only several components of the solution vector), which assumes only the positivity of elements in a corresponding row of the Green's matrix. The main theorem of the paper claims the equivalence of positivity of all elements in the nth row of the Green's matrices of the initial and two other problems, non-oscillation of the nth component of the solution vector and a corresponding assertion about differential inequality of the de La Vallee Poussin type. Necessary and sufficient conditions of the partial monotonicity are obtained. It is demonstrated that our sufficient tests of positivity of the elements in the nth row of the Cauchy matrix are exact in corresponding cases. The main idea in our approach is a construction of an equation for the nth component of the solution vector. In this sense we can say that an analog of the classical Gauss method for solving systems of functional differential equations is proposed in the paper. [ABSTRACT FROM AUTHOR]