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The 8th Asian Pacific Mathematical Olympiad
1996年第8届亚太地区数学奥林匹克
  1. Let ABCD be a quadrilateral AB = BC = CD = DA. Let MN and PQ be two segments perpendicular to the diagonal BD and such that the distance between them is d > BD/2, with M ∈ AD, N ∈ DC, P ∈ AB, and Q ∈ BC. Show that the perimeter of hexagon AMNCQP does not depend on the position of MN and PQ so long as the distance between them remains constant.
  2. Let m and n be positive integers such that n ≤ m. Prove that
    2nn! ≤(m + n)!/(m - n)! ≤ (m2 + m)n
  3. Let P1, P2, P3, P4 be four points on a circle, and let I1 be the incentre of the triangle P2P3P4;
    I2 be the incentre of the triangle P1P3P4; I3 be the incentre of the triangle P1P2P4; I4 be the incentre of the triangle P1P2P3. Prove that I1, I2, I3, I4 are the vertices of a rectangle.
  4. The National Marriage Council wishes to invite n couples to form 17 discussion groups under the following conditions:
    1) All members of a group must be of the same sex; i.e. they are either all male or all female.
    2) The difference in the size of any two groups is 0 or 1.
    3) All groups have at least 1 member.
    4) Each person must belong to one and only one group.
    Find all values of n, n ≤ 1996, for which this is possible. Justify your answer.
  5. Let a, b, c be the lengths of the sides of a triangle. Prove that
    √(a + b - c) + √(b + c - a) + √(c + a - b) ≤ √a + √b + √c ;
    and determine when equality occurs.
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