The 12th All Soviet Union Mathematical Olympiad
1978年第十二届全苏数学奥林匹克 
 a_{n} is the nearest integer to √n. Find 1/a_{1} + 1/a_{2} + ... + 1/a_{1980}.
 ABCD is a quadrilateral. M is a point inside it such that ABMD is a parallelogram. ∠CBM = ∠CDM. Show that ∠ACD = ∠BCM.
 Show that there is no positive integer n for which 1000^{n}  1 divides 1978^{n}  1.
 If P, Q are points in space the point [PQ] is the point on the line PQ on the opposite side of Q to P and the same distance from Q. K_{0} is a set of points in space. Given K_{n} we derive K_{n+1} by adjoining all the points [PQ] with P and Q in K_{n}.
(1) K_{0} contains just two points A and B, a distance 1 apart, what is the smallest n for which K_{n} contains a point whose distance from A is at least 1000 ?
(2) K_{0} consists of three points, each pair a distance 1 apart, find the area of the smallest convex polygon containing K_{n}.
(3) K_{0} consists of four points, forming a regular tetrahedron with volume 1. Let H_{n} be the smallest convex polyhedron containing K_{n}. How many faces does H_{1} have? What is the volume of H_{n} ?
 Two players play a game. There is a heap of m tokens and a heap of n < m tokens. Each player in turn takes one or more tokens from the heap which is larger. The number he takes must be a multiple of the number in the smaller heap. For example, if the heaps are 15 and 4, the first player may take 4, 8 or 12 from the larger heap. The first player to clear a heap wins. Show that if m > 2n, then the first player can always win. Find all k such that if m > kn, then the first player can always win.
 Show that there is an infinite sequence of reals x_{1}, x_{2}, x_{3}, ... such that x_{n} is bounded and for any m > n, we have x_{m}  x_{n} > 1/(m  n).
 Let p(x) = x^{2} + x + 1. Show that for every positive integer n, the numbers n, p(n), p(p(n)), p(p(p(n))), ... are relatively prime.
 Show that for some k, you can find 1978 different sizes of square with all its vertices on the graph of the function y = k sin x.
 The set S_{0} has the single member (5, 19). We derive the set S_{n+1} from S_{n} by adjoining a pair to S_{n}. If S_{n} contains the pair (2a, 2b), then we may adjoin the pair (a, b). If S contains the pair (a, b) we may adjoin (a+1, b+1). If S contains (a, b) and (b, c), then we may adjoin (a, c). Can we obtain (1, 50)? (1, 100)? If We start with (a, b), with a < b, instead of (5, 19), for which n can we obtain (1, n) ?
 An ngon area A is inscribed in a circle radius R. We take a point on each side of the polygon to form another ngon. Show that it has perimeter at least 2A/R .
 Two players play a game by moving a piece on an n x n chessboard. The piece is initially in a corner square. Each player may move the piece to any adjacent square (which shares a side with its current square), except that the piece may never occupy the same square twice. The first player who is unable to move loses. Show that for even n the first player can always win, and for odd n the second player can always win. Who wins if the piece is initially on a square adjacent to the corner ?
 Given a set of n nonintersecting segments in the plane. No two segments lie on the same line. Can we successively add n1 additional segments so that we end up with a single nonintersecting path? Each segment we add must have as its endpoints two existing segment endpoints.
 a and b are positive real numbers. x_{i} are real numbers lying between a and b. Show that (x_{1} + x_{2} + ... + x_{n})(1/x_{1} + 1/x_{2} + ... + 1/x_{n}) ≤ n^{2}(a + b)^{2}/4ab.
 n > 3 is an integer. Let S be the set of lattice points (a, b) with 0 ≤ a, b < n. Show that we can choose n points of S so that no three chosen points are collinear and no four chosen points from a parallelogram.
 Given any tetrahedron, show that we can find two planes such that the areas of the projections of the tetrahedron onto the two planes have ratio at least √2.
 a_{1}, a_{2}, ... , a_{n} are real numbers. Let b_{k} = (a_{1} + a_{2} + ... + a_{k})/k for k = 1, 2, ... , n. Let C = (a_{1}  b_{1})^{2} + (a_{2}  b_{2})^{2} + ... + (a_{n}  b_{n})^{2}, and D = (a_{1}  b_{n})^{2} + (a_{2}  b_{n})^{2} + ... + (a_{n}  b_{n})^{2}. Show that C ≤ D ≤ 2C.
 Let x_{n} = (1 + √2 + √3)^{n}. We may write x_{n} = a_{n} + b_{n}√2 + c_{n}√3 + d_{n}√6 , where a_{n}, b_{n}, c_{n}, d_{n} are integers. Find the limit as n tends to infinity of b_{n}/a_{n}, c_{n}/a_{n}, d_{n}/a_{n }.

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