Program of lessons

Lesson 1.  Arithmetic

Lesson 2.  Algebraic transformations

Lesson 3.  Algebraic equations

Lesson 4.  Logarithmic and exponential equations

Lesson 5.  Inequalities

Lesson 6.  Problems on composition of equations

Lesson 7.  Sequences and progressions

Lesson 8.  Planimetry (Plane geometry)

Lesson 9.  Stereometry (Solid geometry)

Lesson 10.  Trigonometric functions and transformations

Lesson 11.  Trigonometric equations

Lesson 12.  Trigonometric inequalities

Lesson 13.  Vectors and complex numbers

Lesson 14.  Functions and graphs

Lesson 15.  Limits

Lesson 16.  Derivative

Lesson 17.  Integral

Lesson 18.  Sets

Lesson 19.  Theory of combinations and Newton’s binomial

Lesson 20. Principles of probability theory

Lesson 21. Principles of analytic geometry

Lesson 22. Miscellaneous problems


Lesson 1.  Arithmetic

Theory: Whole (natural) numbers. Arithmetical operations. Order of operations. Brackets. Laws of an addition and a multiplication. Divisibility criteria. Prime and composite numbers. Factorization. Resolution into prime factors. Greatest common factor. Least common multiple. Vulgar (simple) fractions. Operations with vulgar fractions. Decimal fractions (decimals). Operations with decimal fractions. Converting a decimal to a vulgar fraction and back. Percents. Ratio and proportion. Proportionality.

Problems: Arithmetic.

Lesson 2.  Algebraic transformations

Theory: Rational numbers. Operations with negative and positive numbers . Monomials and polynomials. Formulas of abridged multiplication. Division of polynomials. Division of polynomial by linear binomial. Divisibility of binomials. Factoring of polynomials. Algebraic fractions. Proportions.

Problems: Algebraic transformations.

Lesson 3.  Algebraic equations

Theory: Equations: common information. Main ways used at solving of equations. Linear equations in one unknown. System of two simultaneous linear equations in two unknowns. System of three simultaneous linear equations in three unknowns. Powers and roots. Arithmetical root. Irrational numbers. Formula of complicated radical. Quadratic equation. Imaginary and complex numbers. Solution of a quadratic equation. Properties of roots of a quadratic equation. Viete's theorem. Factoring of a quadratic trinomial. Equations of higher degrees.

Problems: Algebraic equations.

Lesson 4.  Logarithmic and exponential equations

Theory: Logarithms.

Problems: Logarithmic and exponential equations.

Lesson 5.  Inequalities

Theory: Mathematical induction. Inequalities: common information. Proving and solving of inequalities.

Problems: Inequalities.

Lesson 6.  Problems on composition of equations

Theory: A successful selection of unknowns is the main moment at solving of problems on composition of equations. It is not essential to select as unknowns the values that would be found according to a problem formulation. Sometimes it is advantageous to select some other values as unknowns, to solve the received equations and after this to find the required values using the problem formulation.

Problems: Problems on composition of equations.

Lesson 7.  Sequences and progressions

Theory: Arithmetic and geometric progressions.

Problems: Sequences and progressions.

Lesson 8.  Planimetry (Plane geometry)

Theory: Theorems, axioms, definitions. Straight line, ray, segment. Angles. Parallel straight lines. Euclidean geometry axioms. Polygon. Triangle. Parallelogram and trapezoid. Similarity of plane figures. Similarity criteria of triangles. Geometric locus. Circle and circumference. Inscribed and circumscribed polygons. Regular polygons. Areas of plane figures.

Problems: Plane geometry .

Lesson 9. Stereometry (Solid geometry)

Theory: Common notions. Angles. Projections. Polyhedral angles. Parallelism and perpendicularity of straight lines and planes. Polyhedrons. Prism, parallelepiped, pyramid. Cylinder. Cone. Ball (sphere). Tangent plane of a ball, a cylinder and a cone. Solid angles. Regular polyhedrons. Symmetry. Symmetry of plane figures. Similarity of bodies. Volumes and areas of bodies' surfaces.

Problems: Solid geometry.

Lesson 10.  Trigonometric functions and transformations

Theory: Radian and degree measures of angles . Transforming of degree measure to radian one and back. Trigonometric functions of an acute angle. Solving of right-angled triangles. Relations between trigonometric functions of the same angle. Trigonometric functions of any angle. Reduction formulas. Addition and subtraction formulas. Double-, triple- and half-angle formulas. Transforming of trigonometric expressions to product. Some important correlations. Basic relations between elements of triangle. Solving of oblique triangles. Inverse trigonometric functions. Basic relations for inverse trigonometric functions.

Problems: Trigonometric transformations.

Lesson 11. Trigonometric equations

Theory: Trigonometric equations. Main methods for solving. Systems of simultaneous trigonometric equations.

Problems: Trigonometric equations .

Lesson 12.  Trigonometric inequalities

Theory: Trigonometric inequalities.

Problems: Trigonometric inequalities.

Lesson 13.  Vectors and complex numbers

Theory: Principles of vector calculus. Complex numbers.

Problems: Vectors and complex numbers .

Lesson 14.  Functions and graphs

Theory: Constants and variables. Functional dependence between two variables. Representation of function by formula and table. Designation of functions. Coordinates. Graphical representation of functions. Basic notions and properties of functions. Inverse function. Composite function. Elementary functions and their graphs. Graphical solving of equations. Graphical solving of inequalities.

Problems: Functions and graphs.

Lesson 15.  Limits

Theory: Sequences. Limits of numerical sequences. Some remarkable limits . Limits of functions .

Problems: Limits .

Lesson 16.  Derivative

Theory: Derivative. Geometric and mechanical meaning of derivative. Differential and its relation with derivative. Basic properties of derivatives and differentials. Derivatives of elementary functions. De L'Hospital’s rule . Application of derivative in investigation of functions. Convexity, concavity and inflexion points of a function .

Problems: Derivative.

Lesson 17.  Integral

Theory: Primitive. Indefinite integral. Basic properties of indefinite integral. Integration methods. Some indefinite integrals of elementary functions. Definite integral. Newton-Leibniz formula. Basic properties of definite integral. Geometric and mechanical applications of definite integral. Some definite integrals. Integral with variable upper limit of integration .

Problems: Integral.

Lesson 18.  Sets

Theory: Basic notions. Examples of sets . Operations with sets.

Problems: Sets .

Lesson 19.  Theory of combinations and Newton’s binomial

Theory: Theory of combinations. Newton's binomial.

Problems: Theory of combinations. Newton's binomial.

Lesson 20.  Principles of probability theory

Theory: Events . Definition and basic properties of probability. Conditional probability. Independence of events . Random variables. Characteristics of random variables . Normal (Gaussian) distribution.

Problems: Probability.

Lesson 21.  Principles of analytic geometry

1. Theory (analytic geometry in a plane) : Transformations of coordinates . Straight line. Circle. Ellipse. Hyperbola . Parabola.

2. Theory (analytic geometry in a space) : Transformations of coordinates. Plane. Straight line. Sphere.

Problems: Analytic geometry.

Lesson 22.  Miscellaneous problems

Theory: Any notions from any section of elementary mathematics may be used here.

Problems: Miscellaneous problems .


The each test fulfillment is 3 hours. Solving tests note the "pure" time used to fulfill the test. It's necessary to check will you get through the examination time or not.

Tests: Select test