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 The 38th British Mathematical Olympiad 2002年第38届英国奥林匹克数学竞赛 From the foot of an altitude in an acute-angled triangle perpendiculars are drawn to the other two sides. Show that the distance between their feet is independent of the choice of altitude . n people wish to sit at a round table which has n chairs. The first person takes an seat. The second person sits one place to the right of the first person, the third person sits two places to the right of the second person, the fourth person sits three places to the right of the third person and so on. For which n is this possible ? The real sequence x1, x1, x2, ... is defined by x0 = 1, xn+1 = (3xn + √(5xn2 - 4) )/2. Show that all the terms are integers . S1, S2, ... , Sn are spheres of radius 1 arranged so that each touches exactly two others. P is a point outside all the spheres. Let x1, x2, ... , xn be the distances from P to the n points of contact between two spheres and y1, y2, ... , yn be the lengths of the tangents from P to the spheres. Show that x1x2 ... xn ≥ y1y2 ... yn . 点击此处查看相关视频讲解 在方框内输入单词或词组