DVI exam
DVI is an exam in mathematics at the Moscow State University named after M.V. Lomonosov. The first four problems have a standard level. Problem 5 is advanced level of geometry. Problem 6 is an advanced level equation or inequality. Problem 7 is advanced level of stereometry.
Below are the most difficult problems of this exam in recent years. The headings indicate the year when the problem was used, the variant option of the exam, and the number of the problem.
Contents
- 1 2014 1 Problem 6
- 2 2014 1 Problem 8
- 3 2015 1 Problem 7
- 4 2016 2 Problem 7
- 5 2016 2 Problem 8
- 6 2020 201 problem 6
- 7 2020 202 problem 6
- 8 2020 203 problem 6
- 9 2020 204 problem 6
- 10 2020 205 problem 6
- 11 2020 206 problem 6
- 12 2021 215 problem 7
- 13 2022 221 problem 7
- 14 2022 222 problem 7
- 15 2022 222 problem 6
- 16 2022 224 problem 6
- 17 2023 231 problem 6
- 18 2023 231 EM problem 6
- 19 2023 232 problem 6
- 20 2023 233 problem 6
2014 1 Problem 6
Find all pares of real numbers satisfying the system of equations
Solution
Denote
Denote
is the solution. Let
If
then
if
then
therefore
is the single root.
2014 1 Problem 8
Let
Find and
Solution
where
Answer:
2015 1 Problem 7
A sphere is inscribed in a regular triangular prism with bases Find its radius if the distance between straight lines
and
is equal to
where
and
are points lying on
and
, respectively, and
Solution
The distance from the center of the sphere to the centers of the prism faces is equal to so
In order to find the distance between the lines
and
, one can find the length of two perpendiculars
and
to the line
that are perpendicular to each other. Then
since, when viewed along a straight line
, the segment
is the altitude of a right triangle with legs
and
The plane containe the straight line
The straight line
crossed
at the point
In a right triangle
is the height falling on the hypotenuse,
Let be the projection of
onto plane
Therefore is the projection of
onto plane
at the point
Answer:
2016 2 Problem 7
Let the base of the regular pyramid with vertex be the hexagon
with side
The plane
is parallel to the edge
, perpendicular to the plane
and intersects the edge
at point
so that
The lines along which
intersects the
plane and the base plane are perpendicular.
Find the area of the triangle cut off by the plane from the face
Solution
Denote
are the midpoints of
respectively.
Plane is the plane symmetry of pyramid,
By condition so exist point
is the line along which
intersects the
plane,
is the line along which
intersects the base plane, so
We use the top wiew and get
Denote
and use the side wiew.
Triangle
is the regular triangle with side
, so
Answer: 8.
2016 2 Problem 8
Find the smallest value of the expression
Solution
Denote
The shortest length of a broken line
with fixed ends is equal to the distance between points
and
which is
and is achieved if points
and
are collinear.
Answer:
2020 201 problem 6
Let a triangular prism with a base
be given,
Find the ratio in which the plane
divides the segment
if
Solution
Let be the parallel projections of
on the plane
We use and get
Let
Similarly
Answer:
2020 202 problem 6
Let a tetrahedron be given,
Find the cosine of the angle
between the edges
and
Solution
Let us describe a parallelepiped around a given tetrahedron
and
are equal rectangles.
and
are equal rectangles.
Denote
Answer:
2020 203 problem 6
Let a cube with the base
and side edges
be given. Find the volume of a polyhedron whose vertices are the midpoints of the edges
Solution
Denote the vertices of polyhedron
Triangles
and
are equilateral triangles with sides
and areas
This triangles lies in parallel planes, which are normal to cube diagonal
The distance
between this planes is
So the volume of the regular prism with base
and height
is
Let the area be the quadratic function of
Let
Suppose, we move point
along axis
and cross the solid by plane contains
and normal to axis. Distance from
to each crosspoint this plane with the edge change proportionally position
along axes, so the area is quadratic function from
position.
Answer:
2020 204 problem 6
Let a regular triangular pyramid be given. The circumcenter of the sphere is equidistant from the edge and from the plane of the base of the pyramid. Find the radius of the sphere inscribed in this pyramid if the length of the edge of its base is
Solution
Answer:
2020 205 problem 6
Let the quadrangular pyramid with the base parallelogram
be given.
Point Point
Find the ratio in which the plane divides the volume of the pyramid.
Solution
Let plane cross edge
at point
We make the central projection from point
The images of points
are
respectively.
The image of
is the crosspoint of
and
So lines
and
are crossed at point
Let’s compare volumes of some tetrachedrons, denote the volume of
as
Answer: 1 : 6.
2020 206 problem 6
Given a cube with the base
and side edges
Find the distance between the line passing through the midpoints of the edges
and
and the line passing through the midpoints of the edges
and
Solution
Let points be the midpoints of
respectively. We need to prove that planes
and
are parallel, perpendicular to
Therefore,
Point is the midpoint
For proof we can use one of the following methods:
1. Vectors:
Scalar product
Similarly,
2.
3. Rotating the cube around its axis we find that the point
move to
, then to
then to
Answer:
2021 215 problem 7
The sphere touches all edges of the tetrahedron It is known that the products of the lengths of crossing edges are equal. It is also known that
Find
Solution
The tangent segments from the common point to the sphere are equal.
Let us denote the segments from the vertex to the sphere by
Similarly, we define
If
then
If
The tetrahedron is a regular pyramid with a regular triangle with side
at the base and side edges equal to
Answer: 3.
2022 221 problem 7
The volume of a triangular prism with base
and side edges
is equal to
Find the volume of the tetrahedron
where
is the centroid of the face
is the point of intersection of the medians of
is the midpoint of the edge
and
is the midpoint of the edge
Solution
Let us consider the uniform triangular prism Let
be the midpoint of
be the midpoint of
be the midpoint of
be the midpoint of
The area of
in the sum with the areas of triangles
is half the area of rectangle
so
Denote the distance between these lines
The volume of the tetrahedron is
The volume of the prism is
An arbitrary prism is obtained from a regular one as a result of an affine transformation.
All points on the tetrahedron are defined affinely, which means that the volume ratio will be preserved.
Answer: 5.
2022 222 problem 7
A sphere of diameter is inscribed in a pyramid at the base of which lies a rhombus with an acute angle
and side
Find the angle
if it is known that all lateral faces of the pyramid are inclined to plane of its base at an angle of
Solution 1
Denote rhombus is the vertex of a pyramid
is the center of the sphere,
is the tangent point of
and sphere,
Solution 2
The area of the rhombus
The area of the lateral surface is
Answer:
2022 222 problem 6
Find all possible values of the product if it is known that
and it is true
Solution
Let then for each
equation is true,
Let
no solution.
Answer:
2022 224 problem 6
Find all triples of real numbers in the interval
satisfying the system of equations
Solution
Denote
Similarly,
Therefore
Answer:
2023 231 problem 6
Let positive numbers be such that
Find the maximum value of
Solution
Similarly
Adding this equations, we get:
If
then
Answer:
Explanation for students
For the function under study it is required to find the majorizing function
This function must be a linear combination of the given function
and a constant,
At the supposed extremum point the functions and their derivatives must coincide
2023 231 EM problem 6
Find the maximum value
and all argument values
such that
.
Solution
because
and signs of
and
are different, so
Therefore
2023 232 problem 6
Let positive numbers be such that
Find the maximum value of
Solution
It is clear that
and
Denote
So
If
then
Answer:
2023 233 problem 6
Let positive numbers be such that
Find the maximum value of
Solution
Let Then
Equality is achieved if
Answer: