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The 28th International Mathematical Olympiad Shortlist Problems
1987年第28届国际数学奥林匹克备选题
  1. f is a real-valued function on the reals such that:
    (1) if x ≥ y and f(y) - y ≥ v ≥ f(x) - x, then f(z) = v + z for some z between x and y;
    (2) for some k, f(k) = 0 and if f(h) = 0, then h <= k;
    (3) f(0) = 1;
    (4) f(1987) ≤ 1988;
    (5) f(x) f(y) = f(x f(y) + y f(x) - xy) for all x, y.
    Find f(1987).
    (Australia 6)
  2. S = {a1, a2, ... , an, b1, b2, ... , bn}. There are subsets C1, C2, ... , Ck, such that (1) for no i, j do both ai and bi both belong to Cj, (2) for any pair of distinct elements of S, not of the form ai, bi, there is just one Cj containing both elements. Show that if n > 3, then k ≥ 2n. (USA 3)
  3. Does there exist a polynomial p(x, y) of degree 2 such that, for each non-negative integer n, we have n = p(a, b) for just one pair (a, b) of non-negative integers? (Finland 3)
  4. ABCDA'B'C'D' is any parallelepiped (with ABCD, A'B'C'D' faces and AA', BB', CC', DD' edges). Show that AC + AB'+ AD' <= AB + AD + AA' + AC' (the sum of the thre short diagonals from A is less than the sum of the three edges from A plus the long diagonal from A). (France 5)
  5. Find the smallest real c such that x11/2 + x21/2 + ... + xn1/2 ≤ c (x1 + x2 + ... + xn)1/2 for all n and all real sequences x1, x2, x3, ... which satisfy x1 + x2 + ... + xn ≤ xn+1. (United Kingdom 6)
  6. Show that an/(b+c) + bn/(c+a) + cn/(a+b) ≥ (2/3)n-2 sn-1 for all n ≥ 1, where a, b, c are the sides of a triangle and s = (a + b + c)/2. (Greece 4)
  7. Given any 5 real numbers u0, u1, u2, u3, u4, show that we can always find 5 real numbers v0, v1, v2, v3, v4 such that each ui - vi is integral and ∑i<j(vi - vj)2 < 4. (Netherlands 1)
  8. Does there exist a subset M of Euclidean space such that any plane meets M in a finite non-empty set? (Hungary 1)
  9. Show that for any relatively prime positive integers m, n we can find integers a1, a2, ... , am and b1, b2, ... , bn such that each product aibj gives a different residue mod mn. (Hungary 2)
  10. Two spheres S and S' touch externally and lie inside a cone C. Each sphere touches the cone in a full circle. n solid spheres are arranged in the cone in a ring so that each touches S and S' externally, touches the cone, and touches its two neighbouring solid spheres. What are the possible values of n? (Iceland 3)
  11. Find the number of ways of partitioning {1, 2, 3, ... , n} into three (possibly empty) subsets A, B, C such that (1) for each subset, if the elements are written in ascending order, then they alternate in parity, and (2) if all three subsets are non-empty, then just one of them has its smallest element even. (Poland 1)
  12. ABC is a non-equilateral triangle. Find the locus of the centroid of all equilateral triangles A'B'C' such that A, B', C' are collinear, A', B, C' are collinear, A', B', C are collinear, and both ABC and A'B'C' have their vertices anti-clockwise. (Poland 5)
  13. (German Democratic Republic 2), has been used in the Olympiad.
  14. How many n-digit words can be formed from the alphabet {0, 1, 2, 3, 4} if neighboring digits must differ by exactly 1? (German Federal Republic 1)
  15. (German Federal Republic 2), has been used in the Olympiad.
  16. (German Federal Republic 3),has been used in the Olympiad.
  17. Show that we can color the elements of the set {1, 2, ... , 1987} with 4 colors so that any arithmetic progression with 10 terms in the set is not monochromatic. (Romania 1)
  18. For any positive integer r find the smallest positive integer h(r) such that for any partition of {1, 2, ... , h(r) } into r parts, there are integers a ≥ 0 and 1 ≤ x ≤ y such that a + x, a + y and a + x + y all belong to the same part. (Romania 4)
  19. Given angles A, B, C such that A + B + C < 180o, show that there is a triangle with sides sin A, sin B, sin C and that its area is less than (sin 2A + sin 2B + sin 2C)/8. (USSR 2)
  20. Show that for any integer k > 1, there is an irrational number r such that [rm] = -1 mod k for every natural number m. (Yugoslavia 2)
  21. (USSR 4), has been used in the Olympiad.
  22. (Vietnam 4),has been used in the Olympiad.
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