These were problems given to Jewish candidates to the Mek-mat during
the 70's and 80's. The source for these problems is
[A. Shen, Entrance Examinations to the Mekh-mat,
Mathematical Intelligencer 16 (1994), 6-10].
The problems appear with the year and the name of the examiners as
in Shen's article, corrected in an e-mail of August 8, 1999.
I have written up a complete solution set .
(Lawrentiew, Gnedenko, O.P.~Vinogradov, 1973)
(V.F.~Maksimov, Falunin, 1974)
K is the midpoint of a chord AB.
MN and ST are chords that pass through K. MT intersects
AK at a point P and NS intersects KB at a point Q.
Show that KP=KQ.
(Maksimov, Falunin, 1974)
A quadrangle in space is tangent to a sphere. Show that
the points of tangency are coplanar.
(Nesterenko, 1974)
The faces of a triangular pyramid have the same area.
Show that they are congruent.
(Nesterenko, 1974)
The prime decompositions of different integers m and n
involve the same primes. The integers m+1 and n+1 also
have this property. Is the number of such pairs (m, n)
finite or infinite?
(Podkolzin, 1978)
Draw a straight line that halves the area
and perimeter of a triangle.
(Podkolzin, 1978) Show that
(1/\sin^2 x) \le (1/x^2) + 1- 4/\pi^2 for 0 < x < \pi/2.
(Podkolzin, 1978)
Choose a point on each edge of a tetrahedron.
Show that the volume of at least one of the resulting
tetrahedrons is \le 1/8 of the volume of the initial
tetrahedron.
(Sokolov, Gashkov, 1978)
We are told that a^2 + 4 b^2 = 4, cd = 4. Show
that (a-d)^2 + (b-c)^2 >= 1.6.
(Fedorchuk, 1979; Filimonov, Proshkin, 1980)
We are given a point K on the side AB of a trapezoid
ABCD. Find a point M on the side CD that maximizes
the area of the quadrangle which is the intersection of the
triangles AMB and CDK.
(Pobedrya, Proshkin, 1980)
Can one cut a three-faced angle by a plane so that the
intersection is an equilateral triangle?
(Vavilov, Ugol'nikov, 1981)
Let H_1, H_2, H_3, H_4, be the altitudes of
a triangular pyramid. Let O be an interior point of the
pyramid and let h_1, h_2, h_3, h_4 be the perpendiculars
from O to the faces. Show that
(Vavilov, Ugol'nikov, 1981)
Solve the system of equations y(x+y)^2 = 9,
y (x^3-y^3) = 7.
(Dranishnikov, Savchenko, 1984)
Show that if a,b,c are the sides of
a triangle and A,B,C are its angles, then
(a+b -2c)/sin(C/2)+ (b+c-2a)/sin(A/2)
+(a+c-2b)/sin(B/2) >= 0.
(Dranishnikov, Savchenko, 1984)
In how many ways can one represent a quadrangle as the union
of two triangles?
(Bogatyi, 1984)
Show that the sum of the numbers
1/(n^3 + 3 n^2 + 2n) for n from 1 to 1000 is < 1/4.
(Evtushik, Lyubishkin, 1984)
Solve the equation
x^4 - 14 x^3 + 66 x^2 - 115 x + 66.25 = 0.
(Evtushik, Lyubishkin, 1984)
Can a cube be inscribed in a cone so that 7 vertices
of the cube lie on the surface of the cone?
(Evtushik, Lyubishkin, 1986)
The angle bisectors of the exterior angles A and C of a triangle
ABC intersect at a point of its circumscribed circle. Given
the sides AB and BC, find the radius of the circle.
[From Shen's paper:
``The condition is incorrect: this doesn't happen.'']
(Evtushik, Lyubishkin, 1986)
A regular tetrahedron ABCD with edge a is inscribed
in a cone with a vertex angle of 90 degrees in such a way that
AB is on a generator of the cone. Find the distance from the
vertex of the cone to the straight line CD.
(Smurov, Balsanov, 1986)
Let log(a, b) denote the logarithm of b to base a.
Compare the numbers
log(3, 4) \log(3, 6)... \log (3, 80)
and
2 log(3, 3) \log(3, 5)... log (3, 79)
(Smurov, Balsanov, 1986)
A circle is inscribed in a face of a cube of side a.
Another circle is circumscribed about a neighboring face of the
cube. Find the least distance between points of the circles.
(Andreev, 1987)
Given k segments in a plane, show that
the number of triangles all of whose sides belong to
the given set of segments is less than C k^3/2, for some
positive constant C which is independent of k.
(Kiselev, Ocheretyanskii, 1988)
Use ruler and compasses to construct, from the parabola
y= x^2, the coordinate axes.
(Tatarinov, 1988)
Find all a such that for all x < 0 we have the inequality
ax^2 - 2x > 3a-1.
(Podol'skii, Aliseichik, 1989)
Let A,B,C be the angles and a,b,c the sides of a triangle.
Show that
60 degrees <= (aA + bB + cC)/(a+b+c) <= 90 degrees.