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 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, y-1) or to (x-5, 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 x1 ≥ x2 ≥ x3 ≥ ... satisfies x1 + x4/2 + x9/3 + x16/4 + ... + xN/n ≤ 1 for every square N = n2. Show that it also satisfies x1 + x2/2 + x3 /3 + ... + xn/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 (x2 - y2)1/2 = a(1 - x2 + y2)1/2 and (y - x (x2 - y2)1/2 = b(1 - x2 + y2)1/2 . A set of square carpets have total area 4. Show that they can cover a unit square. xi are real numbers between 0 and 1. Show that (x1 + x2 + ... + xn + 1)2 ≥ 4(x12 + x22 + ... + xn2). 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, ... , (m-1)/m, 1/n, 2/n, ... , (n-1)/n . Given a point P in space and 1979 lines L1, L2, ... , L1979 containing it. No two lines are perpendicular. P1 is a point on L1. Show that we can find a point An on Ln (for n = 2, 3, ... , 1979) such that the following 1979 pairs of lines are all perpendicular: An-1An+1 and Ln for n = 1, ... , 1979. [We regard A-1 as A1979 and A1980 as A1.] Find a sequence a1, a2, ... , a25 of 0s and 1s such that the following sums are all odd: ```a1a1 + a2a2 + ... + a25a25 a1a2 + a2a3 + ... + a24a25 a1a3 + a2a4 + ... + a23a25 ... a1a24 + a2a25 a1a25 ``` 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 在方框内输入单词或词组