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The 12th All Soviet Union Mathematical Olympiad
1978年第十二届全苏数学奥林匹克
  1. an is the nearest integer to √n. Find 1/a1 + 1/a2 + ... + 1/a1980.
  2. ABCD is a quadrilateral. M is a point inside it such that ABMD is a parallelogram. ∠CBM = ∠CDM. Show that ∠ACD = ∠BCM.
  3. Show that there is no positive integer n for which 1000n - 1 divides 1978n - 1.
  4. 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. K0 is a set of points in space. Given Kn we derive Kn+1 by adjoining all the points [PQ] with P and Q in Kn.
    (1) K0 contains just two points A and B, a distance 1 apart, what is the smallest n for which Kn contains a point whose distance from A is at least 1000 ?
    (2) K0 consists of three points, each pair a distance 1 apart, find the area of the smallest convex polygon containing Kn.
    (3) K0 consists of four points, forming a regular tetrahedron with volume 1. Let Hn be the smallest convex polyhedron containing Kn. How many faces does H1 have? What is the volume of Hn ?
  5. 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.
  6. Show that there is an infinite sequence of reals x1, x2, x3, ... such that |xn| is bounded and for any m > n, we have |xm - xn| > 1/(m - n).
  7. Let p(x) = x2 + x + 1. Show that for every positive integer n, the numbers n, p(n), p(p(n)), p(p(p(n))), ... are relatively prime.
  8. 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.
  9. The set S0 has the single member (5, 19). We derive the set Sn+1 from Sn by adjoining a pair to Sn. If Sn 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) ?
  10. An n-gon area A is inscribed in a circle radius R. We take a point on each side of the polygon to form another n-gon. Show that it has perimeter at least 2A/R .
  11. 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 ?
  12. Given a set of n non-intersecting segments in the plane. No two segments lie on the same line. Can we successively add n-1 additional segments so that we end up with a single non-intersecting path? Each segment we add must have as its endpoints two existing segment endpoints.
  13. a and b are positive real numbers. xi are real numbers lying between a and b. Show that (x1 + x2 + ... + xn)(1/x1 + 1/x2 + ... + 1/xn) ≤ n2(a + b)2/4ab.
  14. 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.
  15. 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.
  16. a1, a2, ... , an are real numbers. Let bk = (a1 + a2 + ... + ak)/k for k = 1, 2, ... , n. Let C = (a1 - b1)2 + (a2 - b2)2 + ... + (an - bn)2, and D = (a1 - bn)2 + (a2 - bn)2 + ... + (an - bn)2. Show that C ≤ D ≤ 2C.
  17. Let xn = (1 + √2 + √3)n. We may write xn = an + bn√2 + cn√3 + dn√6 , where an, bn, cn, dn are integers. Find the limit as n tends to infinity of bn/an, cn/an, dn/an .
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