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 The 9th United States of America Mathematics Olympiad 1980年第九届美国数学奥林匹克 A balance has unequal arms and pans of unequal weight. It is used to weigh three objects. The first object balances against a weight A, when placed in the left pan and against a weight a, when placed in the right pan. The corresponding weights for the second object are B and b. The third object balances against a weight C, when placed in the left pan. What is its true weight ? Find the maximum possible number of three term arithmetic progressions in a monotone sequence of n distinct reals . A + B + C is an integral multiple of π. x, y, z are real numbers. If x sin A + y sin B + z sin C = x2 sin 2A + y2 sin 2B + z2 sin 2C = 0, show that xn sin nA + yn sin nB + zn sin nC = 0 for any positive integer n . The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular . If x, y, z are reals such that 0 ≤ x, y, z ≤ 1, show that x/(y + z + 1) + y/(z + x + 1) + z/(x + y + 1) ≤ 1 - (1 - x)(1 - y)(1 - z) . 点击此处查看相关视频讲解 在方框内输入单词或词组