1. Proofs, Proof types (statement, universal and existential quantifier), basic proving methods (direct proof, proof by transposition, proof by contradiction, mathematical induction, pigeonhole principle). Examples, proofs of well-known sentences (Thales' theorem, arithmetic-geometric mean inequality, Cauchy inequality). 2. Sequences. Sequence types, analytic and recurrent definition of sequence. Examples. Series, partial sums and their definition (inductive method with proof, binomial coefficient method). 3. Direct methods of mathematical tasks solving. Method classification (experiment, implication method, equivalent method), selection of appropriate method and its use. Necessity of proof. Examples. 4. Indirect methods of mathematical tasks solving. Method classification (transformation method, complementary method), selection of appropriate method and its use. Examples. 5. Algebraic equations of higher degree. Finding rational roots, usage of Horner scheme, reciprocal polynomials. Examples. System of equations, methods of solving. Parameter in equations and inequalities, roots and their number in dependence on parameter (linear, quadric, logarithmic, exponential equations and inequalities). Examples. 6. Sets of point and their properties. Concept of set of points, basic types of sets in plain and space (circle, perpendicular bisector, angle bisector, sphere, cylinder), methods of sets analyzing (analytic geometry method, inductive method), their use in constructions. 7. Geometric constructions. Classification of construction types, solving methods (algebraic-geometric method), fazes of construction. Examples of triangle and tetragon constructions. 8. Problem of Apollonius. Concept of problem of Apollonius, its different types (point-point-point, point-circle-bisector, ..., circle-circle-circle; special types of problem of Pappus), methods of solving (sets of points, homothety, circle inversion), number of solutions. 9. Word problems. Types of word problems (mathematical, nonmathematical), concrete types of word problems at primary school (distance problems, work problems, mixture problems) + simple optimization problems, solving fazes (mathematization, solving equation/inequality, interpretation). Examples. Thesis of Mathematics Education are described on the website of Department of Mathematics.
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