The 13th All Soviet Union Mathematical Olympiad
1979年第十三届全苏数学奥林匹克 
 T is an isosceles triangle. Another isosceles triangle T' has one vertex on each side of T. What is the smallest possible value of area T'/area T ?
 A grasshopper hops about in the first quadrant (x, y >= 0). From (x, y) it can hop to (x+1, y1) or to (x5, y+7), but it can never leave the first quadrant. Find the set of points (x, y) from which it can never get further than a distance 1000 from the origin.
 In a group of people every person has less than 4 enemies. Assume that A is B's enemy iff B is A's enemy. Show that we can divide the group into two parts, so that each person has at most one enemy in his part.
 Let S be the set {0, 1}. Given any subset of S we may add its arithmetic mean to S (provided it is not already included  S never includes duplicates). Show that by repeating this process we can include the number 1/5 in S. Show that we can eventually include any rational number between 0 and 1.
 The real sequence x_{1} ≥ x_{2} ≥ x_{3} ≥ ... satisfies x_{1} + x_{4}/2 + x_{9}/3 + x_{16}/4 + ... + x_{N}/n ≤ 1 for every square N = n^{2}. Show that it also satisfies x_{1} + x_{2}/2 + x_{3} /3 + ... + x_{n}/n ≤ 3.
 Given a finite set X of points in the plane. S is a set of vectors AB where (A, B) are some pairs of points in X. For every point A the number of vectors AB (starting at A) in S equals the number of vectors CA (ending at A) in S. Show that the sum of the vectors in S is zero.
 What is the smallest number of pieces that can be placed on an 8 x 8 chessboard so that every row, column and diagonal has at least one piece? [A diagonal is any line of squares parallel to one of the two main diagonals, so there are 30 diagonals in all.] What is the smallest number for an n x n board ?
 a and b are real numbers. Find real x and y satisfying: (x  y (x^{2}  y^{2})^{1/2} = a(1  x^{2} + y^{2})^{1/2} and (y  x (x^{2}  y^{2})^{1/2} = b(1  x^{2} + y^{2})^{1/2} .
 A set of square carpets have total area 4. Show that they can cover a unit square.
 x_{i} are real numbers between 0 and 1. Show that (x_{1} + x_{2} + ... + x_{n} + 1)^{2} ≥ 4(x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2}).
 m and n are relatively prime positive integers. The interval [0, 1] is divided into m + n equal subintervals. Show that each part except those at each end contains just one of the numbers 1/m, 2/m, 3/m, ... , (m1)/m, 1/n, 2/n, ... , (n1)/n .
 Given a point P in space and 1979 lines L_{1}, L_{2}, ... , L_{1979} containing it. No two lines are perpendicular. P_{1} is a point on L_{1}. Show that we can find a point A_{n} on L_{n} (for n = 2, 3, ... , 1979) such that the following 1979 pairs of lines are all perpendicular: A_{n1}A_{n+1} and L_{n} for n = 1, ... , 1979. [We regard A_{1} as A_{1979} and A_{1980} as A_{1}.]
 Find a sequence a_{1}, a_{2}, ... , a_{25} of 0s and 1s such that the following sums are all odd:
a_{1}a_{1} + a_{2}a_{2} + ... + a_{25}a_{25}
a_{1}a_{2} + a_{2}a_{3} + ... + a_{24}a_{25}
a_{1}a_{3} + a_{2}a_{4} + ... + a_{23}a_{25}
...
a_{1}a_{24} + a_{2}a_{25}
a_{1}a_{25}
Show that we can find a similar sequence of n terms for some n > 1000.
 A convex quadrilateral is divided by its diagonals into four triangles. The incircles of each of the four are equal. Show that the quadrilateral has all its sides equal.
 对不起，这个题目我没有找到。找到俄文原稿可能还需要一段时间。

点击此处查看相关视频讲解 

