ENTRANCE EXAMINATION IN MATHEMATICS FOR APPLICANTS TO BACHELOR’S PROGRAMMES

Entrance test procedure

1. The entrance test is conducted in accordance with the current Rules of admission to the Bachelor's degree and specialty and the Regulations on the procedure for conducting entrance tests of MIPT.

2. The entrance test in mathematics is conducted with a combination of written and oral forms.

3. The entrance test consists of four parts.

4. The first three parts of the entrance test (lasting 60 minutes each) are a written exam.

5. The first part contains tasks with a short answer and allows for automatic verification. According to the results of the first part, admission to the subsequent parts of the entrance test is carried out. For participants who are not admitted to the subsequent parts, the exam points are awarded based on the results of the first part of the exam.

6. The second and third parts contain tasks with a detailed answer, which are checked by teachers taking the exam (examiners). 

7. The fourth part of the entrance test is an oral conversation, in which the tasks of the first three parts of the exam can also be discussed. The duration of the oral part of the exam is up to 30 minutes.

List of Topics

1. Natural numbers. Divisibility. Prime numbers and composite numbers. Divisibility rules. Greatest common factor and least common multiple.

2. Integer, rational, real numbers and operations on them.

3. Transformation of arithmetic and algebraic expressions. Short multiplication formulas.

4. Numerical inequalities and their properties.

5. Function. Domain and range of a function. Graph of a function. Parity, oddness, periodicity of functions. Linear, quadratic, power, rational functions and their properties.

6. Linear equations. Quadratic equations. Rational equations. Module equations. Equations of higher degrees. Factorization of polynomials.

7. Linear inequalities. Square inequalities. Rational inequalities. Absolute-value inequalities.

8. Root of a number and its properties. Arithmetic root. Irrational equations. Irrational inequalities.

9. Arithmetic and geometric progressions and their properties.

10. Combinatorics. Sum and product rules. Permutations, variations, combinations.

11. Formation & solving equations (interest, mixture, motion, job, etc).

12. Trigonometric formulas. Trigonometric and inverse trigonometric functions and their properties. Conversion of trigonometric expressions. Trigonometric equations and inequalities.

13. Properties of exponents. Logarithms and their properties. Exponential and logarithmic functions and their properties. Exponential and logarithmic equations and inequalities.

14. Derivative of a function. Study of functions using derivatives.

15. Problem with Parameter Values

16. Systems of equations and inequalities.

17. Points on the coordinate plane.

18. Plane geometry:

- adjacent and vertical angles,

- signs and properties of isosceles triangle; SSS, SAS, ASA and AAS rules;

- theorems of parallel lines, sum of angles of a triangle, angle sum of polygons;

- locus of points (a set of internal points of an angle equidistant from its sides, a set of points equidistant from the endpoints of a segment);

- medians, altitudes, angle bisectors of a triangle and their properties;

- similarity of triangles, the Intercept theorem, the proportional segments theorem;

- quadrilaterals; parallelogram, rectangle, rhombus, square, trapezoid and their properties;

- proportional segments in a right triangle, the Pythagorean theorem;

- area and its properties;

- area formulas: triangle, parallelogram, trapezoid;

- the point of concurrency of three altitudes, medians, angle bisectors, perpendicular bisectors of a triangle

- lows of sines, cosines, the Menelaus' theorem for a triangle;

- circle and its properties;

- tangent to a circle and its properties;

- intersecting chord theorem;

- angles in a circle theorems (inscribed angle, central angle, angle between a tangent and a chord);

- circumcircle of a triangle, inscribed circle of a triangle;

- quadrilateral circumscribing a circle, quadrilateral inscribed in a circle;

- regular polygons and their properties;

- length of circumference of a circle, area of a circle and its parts,

- vectors, dot product;

- the method of coordinates on a plane.

19. Solid geometry. Parallel and perpendicular lines and planes. Surface area and volume of a three dimensional figure. Cube, parallelepiped, prism, pyramid, ball, cylinder, cone and their properties. Vectors and coordinates in the space. Cross sections of polyhedra. Angles and distances in the space.