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Online math courses (and examples of self-test problems) for high school and admission to university/college

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You can enroll in our online courses at any time:

  1. Please contact us via chat on this website to pre-agree a planned start date of your classes for specific online course.
  2. Immediately after prior agreement do purchase specific agreed online courses.
  3. After your purchase, we will schedule the start of your classes as previously agreed and give you all the necessary instructions.

Refer to "Purchases and Payments" section (link) of the "Terms and Conditions" (link).

Refreshing knowledge and skills:

If refreshing is requested, we assume that you already have good understanding of the given subject matter and will focus to refresh your knowledge as quick as possible. You will receive same or more concise instruction and less time-consuming, but more advanced exercises. This will allow you to proceed at a times faster pace and is therefore recommended when there are tight deadlines.

Learn with personal tutor:

If learning with personal tutor is requested, we ensure that the same dedicated tutor will closely monitor your progress in the specified online course or courses for which the personal tutor was requested. This provides more accurate, student-specific, highly personalized tutoring: instruction, motivation, clarification of concepts and assistance to reinforce learning. However, personal tutoring may proceed at a slower pace due to scheduling challenges that we face then and is therefore recommended only when there are no tight deadlines.

Individual or group courses

You can enroll in any of our online courses with your own group of students (e.g. your friends, classmates, just random study mates, etc.) to take that course together, which allows to discount per-student price by almost half. The larger your group, the lower the price per student, and even a group of just 2 students (i.e. you and someone else) allows you to apply a significant discount. For any course, the page you are currently viewing displays the single-student (i.e. individual course) price, as well as the per-student price of the same course for your group of, at the moment, 3 students. Instead, get this page with prices for your group of any (from 2 to 10) number of students:

Commonly used elements of mathematical logic and set theory

You will know:

  • Logical statements and operations on them.
  • Logical predicates.
  • Domain of truth of a logical predicate.
  • Logical operations on predicates.
  • Logical quantifiers.
  • Theorems and their types.
  • Common structure of a theory.
  • Sets and operations on them.
  • Euler diagrams.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:1

Find volume of the pyramid from its side face

AD is the height of the triangular pyramid ABCD that has BC = 1, CD = 2, ∠BCD = 120° and dihedral ∠BC = 45°. Find the volume of the pyramid.

Ref:6

Systematic long-term Math preparation for university/college admission and high school final exams (30% off)

You will know:

  • Commonly used elements of mathematical logic and set theory
  • Real numbers (natural, rational, and irrational) and operations with them
  • Ratios and proportions. Percentages. Basic percentage math problems. Math problems in natural language.
  • Exponentiation and roots of a number
  • Algebraic expressions and their transformations
  • Eequation and system of equations
  • Inequality and system of inequalities
  • Comparison of real numbers given by algebraic expressions
  • Functions
  • Exponential and logarithmic functions and expressions
  • Exponential and logarithmic equations and their systems
  • Exponential and logarithmic inequalities and their systems
  • Trigonometric functions and expressions
  • Trigonometric equations and their systems
  • Trigonometric inequalities and their systems
  • Numerical sequences
  • Limit and continuity of a numeric function
  • Derivative of a numeric function and Its applications
  • Integral and its applications
  • Complex numbers and polynomials
  • Combinatorics, probability theory and mathematical statistics
  • Geometry on the plane (planimetry): core definitions, axioms and theorems
  • Triangle
  • Quadrilateral
  • Polygon
  • Coordinates and vectors in the plane
  • Coordinates and vectors in the plane (advanced level)
  • Geometric transformations
  • Straight lines and planes in space
  • Lines and planes in space (advanced level)
  • Polyhedra, rotational bodies
  • Polyhedra, rotational bodies (advanced level)
  • Coordinates and vectors in space

The 30% discount of this Course applies to the total cost of the Bundle of courses that consists of the Courses (as if purchased individually) covered by this Course. Any Course that have been started, including this Course, is not refundable, see our Terms and Conditions.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:2

Find the perimeter of a triangle with a bisector

The bisector of a triangle divides its side into segments of length 28 and 12, and the difference of the other two sides is 18. Find the perimeter of the triangle.

Ref:5

Real numbers (natural, rational, and irrational) and operations with them

You will know:

  • Natural, integer, rational, irrational, real numbers.
  • Number notations and conversion between them.
  • Properties of adding and multiplying operations with real numbers.
  • The test criteria/rules/traits (i.e. сharacteristics) of divisibility by 2, 3, 5, 9, 10 for integer numbers.
  • Prime numbers. Factoring a natural number into prime factors.
  • Rules for finding the greatest common divisor and least common multiple of numbers.
  • Rules for rounding decimal fractions.
  • Number intervals.
  • Absolute value of a real number and its properties.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:3

log(a, x) = (2/3)*log(a, b) - (1/3)*log(a, c)

Solve the equation for x. Simplify the solution.

Ref:4

Custom math preparation course for university/college admission and high school final exams (20% off, if above 500 USD)

You will know:

  • Custom math preparation topics tailored to individual needs.

You can arrange for your custom Course by contacting client support via website chat.

The 20% discount of the custom Course applies to the total cost of the Bundle of courses that consists of the Courses (as if purchased individually) covered by Custom Course, if that total cost exceeds 500 USD. Any Course that have been started, including custom Course, is not refundable, see our Terms and Conditions.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:4

Find angles of triangle 3, 4, 5

The lengths of the triangle's sides are: 3, 4, 5. Find all the angles.

Ref:1

Ratios and proportions. Percentages. Basic percentage math problems. Math problems in natural language.

You will know:

  • Numeric fractions (ratios). Basic property of a numeric fraction. Reduction and simplification of numeric fraction. Finding common denominator of two numeric fractions.
  • Proportions. Basic property of a proportion.
  • Arithmetic operations on rational numbers.
  • Comparison of rational numbers.
  • The quotient and remainder of dividing one natural number by another.
  • Definition of percentage.
  • Rules for performing percentage calculations.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:5

Find angles of triangle 2, 3, 4

The lengths of the triangle's sides are: 2, 3, 4. Find all the angles.

Ref:2

Exponentiation and roots of a number

You will know:

  • Definition and properties of a n-th root and arithmetic root of a n-th degree.
  • Definition and properties of an exponentiation with any real (natural, integer, rational and irrational) exponents.
  • Comparison of irrational numbers. Comparison of irrational numbers derived from roots.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:6

Find angles of triangle 1, 2, 3

The lengths of the triangle's sides are: 1, 2, 3. Find all the angles.

Ref:3

Algebraic expressions and their transformations

You will know:

  • Algebraic operations.
  • Algebraic expressions. Arithmetic expressions. Variables and constants in expressions.
  • Precedence and associativity of algebraic operations in expressions.
  • Definition of identically equal expressions, identical transformation of expression, identity.
  • Definition of a monomial and a polynomial.
  • Rules for adding, subtracting, and multiplying monomials and polynomials.
  • Short multiplication formulas.
  • Factoring a polynomial.
  • Definition of a fractional rational expression.
  • Rules for performing operations with fractional rational algebraic expressions.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:7

Eequation and system of equations

You will know:

  • Definition of equation and system of equations.
  • General methods and techniques (factoring, variable substitution, application of properties and graphs of functions) for solving equations and their systems.
  • How to solve polinomial algebraic equations: linear, quadratic, higher degrees.
  • How to solve fractional rational algebraic equations.
  • How to solve irrational algebraic equations.
  • How to solve systems of algebraic equations.
  • Solving word problems by converting them to equations and systems of equations.
  • How to analyze and investigate (parametric) equations, their systems depending on coefficients.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:8

Inequality and system of inequalities

You will know:

  • Definition of inequality and system of inequalities.
  • General methods and techniques (factoring, method of intervals, variable substitution, application of properties and graphs of functions, etc.) for solving inequalities and their systems.
  • How to solve polinomial algebraic inequalities: linear, quadratic, higher degrees.
  • How to solve fractional rational algebraic inequalities.
  • How to solve irrational algebraic inequalities.
  • How to solve systems of algebraic inequalities.
  • Solving word problems by converting them to inequalities and systems of inequalities.
  • How to analyze and investigate (parametric) inequalities, their systems depending on coefficients.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:9

Comparison of real numbers given by algebraic expressions

You will know:

  • How to compare real numbers given by algebraic expressions.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:10

Functions

You will know:

  • Concept of a function: argument, value, (natural) domain of definition (pre-image), image (range of values), domain of values (co-domain), graphs.
  • Properties of numerical functions: odd, even, periodic, etc.
  • Algebraic functions: power functions, polynomial functions (linear, quadratic, higher degrees), fractional rational algebraic functions, irrational functions.
  • Transformation of function graphs.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:11

Exponential and logarithmic functions and expressions

You will know:

  • Definition and properties of exponential function.
  • Definition and properties of logarithm.
  • Basic logarithmic identity.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:12

Exponential and logarithmic equations and their systems

You will know:

  • Definition of exponential and logarithmic equation and system of equations.
  • How to solve exponential equations.
  • How to solve logarithmic equations.
  • How to solve systems of exponential and logarithmic equations.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:13

Exponential and logarithmic inequalities and their systems

You will know:

  • Definition of exponential and logarithmic inequalities and system of inequalities.
  • How to solve exponential inequalities.
  • How to solve logarithmic inequalities.
  • How to solve systems of exponential and logarithmic inequalities.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:14

Trigonometric functions and expressions

You will know:

  • Definition of trigonometric functions: sine, cosine, tangent.
  • Basic relationships between trigonometric functions.
  • Formulas for lowering the power
  • Addition formulas and their consequences.
  • Other trigonometric identities with trigonometric functions.
  • Definition of inverse trigonometric functions: arcsine, arccosine, arctangent.
  • Basic relationships between inverse trigonometric functions.
  • Other trigonometric identities with inverse trigonometric functions.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:15

Trigonometric equations and their systems

You will know:

  • Definition of trigonometric equation and system of equations.
  • Solutions of simplest trigonometric equations.
  • How to solve trigonometric equations.
  • How to solve systems of trigonometric equations.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:16

Trigonometric inequalities and their systems

You will know:

  • Definition of trigonometric inequality and system of inequalities.
  • Solutions of simplest trigonometric inequalities.
  • How to solve trigonometric inequalities.
  • How to solve systems of trigonometric inequalities.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:17

Numerical sequences

You will know:

  • Numerical sequences as functions of a natural argument. Methods of defining sequences. Important classes of numerical sequences (monotonic, bounded, etc.).
  • Limit of a numerical sequence. Geometric interpretation of the limit of a numerical sequence.
  • Fundamental theorems about the limit of numerical sequence.
  • Mathematical constant e.
  • Length of a circle and area of a circle. Mathematical constant π.
  • Arithmetic and geometric progressions: definitions, formulas, and sums.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:18

Limit and continuity of a numeric function

You will know:

  • Limit of a numeric function at a point.
  • Fundamental theorems about the limit of numeric function at a point.
  • Continuity of a function at a point and on an interval. Properties of continuous functions.
  • Points of discontinuity of a function.
  • Concept of a numeric function's limit at infinity and infinitely large function at a point. Vertical and horizontal asymptotes of a numeric function's graph.
  • "Beautiful limits".

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:19

Derivative of a numeric function and Its applications

You will know:

  • Derivative of a numeric function: definition, geometric and physical meaning/interpretation. Mathematical problems leading to the concept of a derivative function.
  • Derivatives of power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
  • Rules for finding derivatives: sum, product and quotient of functions.
  • Derivative of a composite function and inverse function.
  • Finding slope, angle of inclination and equation of the tangent line to the graph of a function.
  • Criterion of function constancy. Conditions for function increasing and decreasing. Extrema of a function. Maximum and minimum values of a function on an interval.
  • Investigation of functions using derivatives: analysis of monotonicity, extrema, and graphical representation of functions.
  • Applying the derivative to prove identities and inequalities, as well as to solve equations and inequalities.
  • Application of derivatives to solving problems.
  • Higher-order derivatives.
  • Concept of convexity of a function and inflection points. Finding intervals of convexity of a function and its inflection points.
  • Applying the first and second derivatives to analyze functions and construct their graphs.
  • Asymptotes of a numeric function's graph.
  • [Jensen's inequality and its application.]

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:20

Integral and its applications

You will know:

  • Antiderivative (indefinite integral) and its properties.
  • Methods of finding antiderivative (indefinite integral).
  • Definite integral, its physical and geometric meaning.
  • Computing the definite integral.
  • Examples of problems leading to the concept of a definite integral. Using definite integral to solve problems.
  • Computing the areas of plane figures.
  • Computing volumes of solids.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:21

Complex numbers and polynomials

You will know:

  • Definition of a complex number and the set of all complex numbers.
  • Geometric interpretation of a complex number.
  • Algebraic and trigonometric forms of expressing a complex number.
  • Operations on complex numbers in various forms of expression.
  • De Moivre's formula.
  • n-th root of a complex number.
  • Polynomial and its roots.
  • Factoring a polynomial into irreducible factors.
  • Multiplicity of the root.
  • Fundamental theorem of algebra.
  • Polynomial of the third degree.
  • Equations of higher degrees.
  • Cardano's formula.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:22

Combinatorics, probability theory and mathematical statistics

You will know:

  • Permutations, combinations, arrangements: definitions, rules.
  • Probability of random events: classical definition, combinatoric schemes.
  • Statistical data characteristics: sample range, mode, median, mean.
  • Presentation of statistical data: graphical, tabular, textual forms.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:23

Geometry on the plane (planimetry): core definitions, axioms and theorems

You will know:

  • Elementary geometric figures: points, straight lines, rays, segments, angles.
  • Axioms of planimetry.
  • Angle measurement, angle calculation.
  • Properties of adjacent, vertical, parallel, and perpendicular angles.
  • Properties of parallel and perpendicular straight lines.
  • Distance between parallel straight lines. Middle perpendicular (perpendicular bisector) of a segment. Distance from point to straight line.
  • Conditions of parallelism: Thales' theorem, its generalization.
  • Circles: definitions, properties of central and inscribed angles, tangent properties.
  • Formulas for calculating the length of a circle and its arcs.
  • Formulas for calculating the area of disk (round), sector and segment of disk (round).

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:24

Triangle

You will know:

  • Triangles and their basic properties. Classifying triangles by sides and angles.
  • Triangle inequality.
  • Triangle angle sum theorem.
  • Equality criteria for triangles.
  • Medians, bisectors, and altitudes of a triangle and their properties.
  • Midline of a triangle and its properties.
  • Circumscribed and inscribed triangles. Circumscribed (circumcircle) and inscribed (incircle) circle.
  • Relationship between triangle sides and angles.
  • Theorem of cosines. Theorem of sines.
  • Geometric similarity of triangles and similarity criteria.
  • Formulas for calculating the area of triangles.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:25

Quadrilateral

You will know:

  • Quadrilateral, its definition, elements and properties.
  • Parallelogram, its definition and properties.
  • Rectangle, rhombus, square, their definitions and properties.
  • Trapezoid, midline of a trapezoid, their definitions and properties.
  • Quadrilaterals inscribed into and circumscribed around a circle.
  • Sum of angles in a quadrilateral.
  • Formulas for calculating the area of squares, parallelograms, rhombuses and trapezoids.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:26

Polygon

You will know:

  • Polygon, its definition, elements and properties.
  • Regular polygon and its properties.
  • Polygons inscribed into and circumscribed around a circle.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:27

Coordinates and vectors in the plane

You will know:

  • Rectangular coordinate system on the plane, point coordinates; distance between two points.
  • Finding the coordinates of the midpoint of a segment and the formula for calculating the distance between two points.
  • Equations of a line and a circle.
  • Concept of a vector, zero vector, vector modulus.
  • Collinear vectors, opposite vectors, equal vectors.
  • Vector coordinates.
  • Addition and subtraction of vectors, multiplication of a vector and number.
  • Angle between vectors.
  • Scalar product of vectors.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:28

Coordinates and vectors in the plane (advanced level)

You will know:

  • Decomposition of a vector into two non-collinear vectors.
  • Properties of scalar product of vectors.
  • Formula for finding the angle between vectors given their coordinates.
  • Conditions for collinearity and perpendicularity of vectors given their coordinates.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:29

Geometric transformations

You will know:

  • Definitions of basic types of geometric transformations on the plane: translation, symmetry with respect to a point and a line, rotation, parallel transfer.
  • Equality of geometric figures.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:30

Straight lines and planes in space

You will know:

  • Core definitions of stereometry.
  • Core axioms and theorems of stereometry.
  • Relative position of: straight lines in space, straight line and plane in space, two geometric planes.
  • Parallelism of: straight lines in space, straight line and plane in space, two geometric planes.
  • Orthogonal projection.
  • Perpendicularity of: straight lines in space, straight line and plane in space, two geometric planes.
  • Theorem of three perpendiculars.
  • Distance: from a point to a plane, from a straight line to a plane parallel to it, between parallel planes.
  • Angle between: straight lines in space, straight line and plane, planes.
  • Dihedral angle. Linear angle of a dihedral angle.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:31

Lines and planes in space (advanced level)

You will know:

  • Mutually skew (non-parallel and non-intersecting) straight lines in space.
  • Orthogonal projection.
  • Distance between two mutually skew lines in space.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:32

Polyhedra, rotational bodies

You will know:

  • Polyhedra, its definition, elements and properties.
  • Basic types of polyhedra: prism, parallelepiped, pyramid. Unfolding of a prism and a pyramid.
  • Rotational bodies, basic types of rotational bodies: cylinder, cone, sphere.
  • Sections of polyhedra.
  • Sections of a cylinder and a cone: axial sections, sections by planes parallel to their bases.
  • Section of a sphere by a plane.
  • Formulas for calculating surface areas and volumes of a prism and a pyramid.
  • Formulas for calculating surface areas and volumes of a cylinder, cone, sphere.
  • Formula for calculating the area of a sphere.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:33

Polyhedra, rotational bodies (advanced level)

You will know:

  • Truncated pyramid.
  • Truncated cone (conoid).

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:34

Coordinates and vectors in space

You will know:

  • Rectangular coordinate system in space, point coordinates.
  • Formula for calculating the distance between two points and the coordinates of the midpoint of a segment.
  • Concept of a vector, vector modulus, collinear vectors, equal vectors, vector coordinates.
  • Addition, subtraction of vectors, multiplication of a vector and number.
  • Scalar product of vectors and its algebraic properties.
  • Angle between vectors.
  • Symmetry with respect to the zero coordinates point and coordinate planes.
  • Vector product of vectors and its algebraic properties.

You can now either:

Alternatively, with your group of 3 students (change that number, if needed) you can now either:

Ref:35

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