COURSE AIMS AND OBJECTIVES: The aim of seminar is to qualify students to independent and project (collaborator  group) work: investigation, discovering and search on the literature (using all available mediums  printed and electronic, especially on Internet), to preparing seminar lectures in the written form (by using computers) and oral lecturing (presentation) on the given subject. Particularly, students will get introduced with the subjects from elementary mathematics assigned to, first of all, work with the mathematics talented students in primary and secondary schools  mathematic groups and preparations for the mathematical competitions. The topics of student projects will be chosen in a way to enable students, future mathematic teachers, to recognition of individuality and opening the students to questions, as the fundamental principles in mathematics teaching on all educational levels.
COURSE DESCRIPTION AND SYLLABUS:
Seminar  Mathematical competitions will be realized as a project education, estimating all its levels and lawfulness. Each project teams will be composed of 3 students, and the project problem will be to elaborate one of the given problems and to present it in written and oral form to other visitors of Seminar. The subjects are as follows:
1. Number theory. Divisible, congruence, Chinese remainder theorem. Sistems of linear equations. Fermat's and Euler's theorem. Diophantine equations, homogeneous sistems of linear diophantine equations.
2. Mathematical induction. The history of mathematical induction. Application in arithmetic, algebra, trigonometry, analysis and geometry.
3. Dirichlet's princip. Basic examples. Application in number theory. Application in geometry.
4. Complex numbers. Trigonometry form of complex number. De Moivre's theorem. Polynomials and complex numbers. Application in algebra. Application in geometry. Application in (analytic) geometry.
5. Combinatorics and probability theory. Enumeration, binomial coefficients, Pascal triangle. Inclusion  exclusion principle. Recursive relations. Relative probability, Bayes's theorem.
6. Inequalities. Algebraic inequalities. Geometric inequalities.
7. Planimetry. Calculation problems. Proving problems. Geometric locus of points. Constructions. The problems to determine minimum and maximum.
8. Stereometry. Calculation problems. Prooving problems. Geometric locus of points. Constructions. The problems to determine minimum and maximum.
9. Analitical geometry. The equations of the line, hyperbola, ellipse and parabola. Application to geometric problems.
10. Vectors. Application of vectors in the plain and in the space. Linear dependance and independance of vectors. Scalar and vector product.
11. Trigonometry. Trigonometric equations and sistem of equations. Trigonometric inequalities. Determining different trigonometric summs.
12. Ceva's theorem and Menelaus' theorem. Applications to different plane problems. Applications to different solid problems.
The activities will be organized in the following stages:
1. Establishing the students projective teams and choosing the projective topics. (1 week)
2. Work on projects. (3 weeks) Each project team work independantly on realization of the project (exploring, researching the literature, discovering and selecting the information, preparing seminar in a written forme and preparing a public presentation of particular theme). In the terms of teaching, meeting of projective teams and teachers  the bearer of subject will be helded, the teams will inform about their work on the project, and the teacher (a menager of all projects) will orient and assist them.
3. Public presentation of the projectives themes. (11 weeks) Every projective team will present the results of their research to other visitors of the seminar.
