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,

- (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**H_1^4+H_2^4 + H_3^4 + H_4 ^4 >= 1024 h_1 h_2 h_3 h_4.* - (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.