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The 9th Asian Pacific Mathematical Olympiad
1997年第9届亚太地区数学奥林匹克
  1. Given S = 1 + 1/(1 + 1/3)+1/(1 + 1/3 + 1/6) + ... + 1/ (1 + 1/3 + 1/6 + ... + 1/1993006),
    where the denominators contain partial sums of the sequence of reciprocals of triangular
    numbers (i.e. k = n(n + 1)=2 for n = 1, 2, ... , 1996). Prove that S > 1001.
  2. Find an integer n, where 100 ≤ n ≤ 1997, such that (2^n + 2)/n is also an integer.
  3. Let ABC be a triangle inscribed in a circle and let la =ma/Ma, lb =mb/Mb, lc =mc/Mc,
    where ma, mb, mc are the lengths of the angle bisectors (internal to the triangle) and Ma,Mb, Mc are the lengths of the angle bisectors extended until they meet the circle. Prove that
    la/sin2 A + lb/sin2 B + lcsin2 C ≥ 3 .
  4. Triangle A1A2A3 has a right angle at A3 . A sequence of points is now defined by the following iterative process, where n is a positive integer. From An (n ≥ 3), a perpendicular line is drawn to meet An-2An-1 at An+1.
    (a) Prove that if this process is continued indefinitely, then one and only one point P is interior to every triangle An-2An-1An, n ≥ 3.
    (b) Let A1 and A3 be fixed points. By considering all possible locations of A2 on the plane, find the locus of P.
  5. Suppose that n people A1, A2, ... , An, (n ≥ 3) are seated in a circle and that Ai has ai objects such that
              a1 + a2 + ... + an = nN
    where N is a positive integer. In order that each person has the same number of objects, each
    person Ai is to give or to receive a certain number of objects to or from its two neighbours
    Ai-1 and Ai+1. (Here An+1 means A1 and An means A0.) How should this redistribution be performed so that the total number of objects transferred is minimum ?
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