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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 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. Let m and n be positive integers such that n ≤ m. Prove that 2nn! ≤(m + n)!/(m - n)! ≤ (m2 + m)n 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. 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. 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. 点击此处查看相关视频讲解 在方框内输入单词或词组