奥数之家 奥数论坛 简短留言
 | 首页 | 竞赛大纲 | 优秀前辈 | 视频提示 | 专题讲座 | 论文锦集 | 综合训练 | 修身养性 | 家教平台 | 奥数论坛 |
 The Seventeenth British Mathematical Olympiad 1981年第十七届英国奥林匹克数学竞赛 ABC is a triangle. Three lines divide the triangle into four triangles and three pentagons. One of the triangle has its three sides along the new lines, the others each have just two sides along the new lines. If all four triangles are congruent, find the area of each in terms of the area of ABC . An axis of a solid is a straight line joining two points on its boundary such that a rotation about the line through an angle greater than 0 deg and less than 360 deg brings the solid into coincidence with itself. How many such axes does a cube have? For each axis indicate the minimum angle of rotation and how the vertices are permuted . Find all real solutions to x2y2 + x2z2 = axyz, y2z2 + y2x2 = bxyz, z2x2 + z2y2 = cxyz, where a, b, c are fixed reals . Find the remainder on dividing x81 + x49 + x25 + x9 + x by x3 - x. The sequence u0, u1, u2, ... is defined by u0 = 2, u1 = 5, un+1un-1 - un2 = 6n-1. Show that all terms of the sequence are integral . Show that for rational c, the equation x3 - 3cx2 - 3x + c = 0 has at most one rational root. If x and y are non-negative integers, show that there are non-negative integers a, b, c, d such that x = a + 2b + 3c + 7d, y = b + 2c + 5d iff 5x ≥ 7y . 点击此处查看相关视频讲解 在方框内输入单词或词组