**
Program of lessons
**

**
**

**
Lesson 2. Algebraic transformations
**

**
Lesson 4. Logarithmic and exponential equations
**

**
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 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
**

**
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
.
**

**
Tests
**

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
**