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The 42nd Vietnam Mathematical Olympiad
2003年第42届越南数学奥林匹克
  1. Let us consider a set S = {a1 < a2 < ... < a2004}, satisfying the following properties: f(ai) < 2003 and f(ai) = f(aj) i,j ∈ { 1 , 2 , ... , 2004 }, where f(ai) denotes the number of elements (in S) which are relatively prime to ai. Find the least positive integer k for which every k-ubset of S, having the above mentioned properties and there are two distinct elements with greatest common divisor greater than 1.
  2. Find all real value of a, for which there exists one and only one function f : RR and satisfying the equation:
           f(x2 + t + f(y)) = f2(x) + a×y
    for all x ,y ∈ R .
  3. In the plane there two circles ○1,○2 intersecting each other at two points A and B. Tangents
    of ○1 at A and B meet at K. Let us consider an arbitrary point M (which is different from A and B) on ○1. The line MA meets ○2 again at P. The line MK meets ○1 again at C. The line CA meets ○2 again at Q. Show that the midpoint of PQ lies on the line MC and the line PQ passes a fixed point when the point M moves on ○1.
  4. Let {xn} n = 1, 2, ... be a given sequence defined by: x1 = 603; x2 = 102 and xn+2 = xn+1 + xn + √(xn+1·xn - 2) n ≥ 1. Show that:
    1) xn is a positive integer for any n ≥ 1;
    2) There are in finitely many positive integer n for which the decimal representation of xn ends with 2003;
    3) There is no positive integer n for which the decimal representation of xn ends with 2004.
  5. Let us consider a convex hexagon ABCDEF. Let A1,B1,C1,D1,E1,F1 be midpoint of the side AB,BC,CD,DE,EF, FA respectively. Denote by p and p1, respectively, the perimeter of the hexagon ABCDEF and the hexagon A1B1C1D1E1F1. Suppose that all inner angles of hexagon A1B1C1D1E1F1 are equal. Prove that: p ≥ (2√3/3)p1, when does the equality hold ?
  6. Let S be a set of positive integers in which the greatest and smallest elements are relatively prime. For natural number n, let Sn denote the set of natural numbers which can be represented as sum of at most n elements(not necessary different) in S. Let a be greatest element in S. Prove that there are positive integers k and b such that |Sn| = a·n + b, n > k.
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