Mekh-mat entrance examination problems

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 .

  1. (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.

  2. (Maksimov, Falunin, 1974) A quadrangle in space is tangent to a sphere. Show that the points of tangency are coplanar.

  3. (Nesterenko, 1974) The faces of a triangular pyramid have the same area. Show that they are congruent.

  4. (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?

  5. (Podkolzin, 1978) Draw a straight line that halves the area and perimeter of a triangle.

  6. (Podkolzin, 1978) Show that (1/\sin^2 x) \le (1/x^2) + 1- 4/\pi^2 for 0 < x < \pi/2.

  7. (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.

  8. (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.

  9. (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.

  10. (Pobedrya, Proshkin, 1980) Can one cut a three-faced angle by a plane so that the intersection is an equilateral triangle?

  11. (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.

  12. (Vavilov, Ugol'nikov, 1981) Solve the system of equations y(x+y)^2 = 9, y (x^3-y^3) = 7.

  13. (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.

  14. (Dranishnikov, Savchenko, 1984) In how many ways can one represent a quadrangle as the union of two triangles?

  15. (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.

  16. (Evtushik, Lyubishkin, 1984) Solve the equation x^4 - 14 x^3 + 66 x^2 - 115 x + 66.25 = 0.

  17. (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?

  18. (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.'']

  19. (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.

  20. (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)

  21. (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.

  22. (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.

  23. (Kiselev, Ocheretyanskii, 1988) Use ruler and compasses to construct, from the parabola y= x^2, the coordinate axes.

  24. (Tatarinov, 1988) Find all a such that for all x < 0 we have the inequality
    ax^2 - 2x > 3a-1.

  25. (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.
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