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 8th United States of America Mathematics Olympiad 1979年第八届美国数学奥林匹克 Find all sets of 14 or less fourth powers which sum to 1599. N is the north pole. A and B are points on a great circle through N equidistant from N. C is a point on the equator. Show that the great circle through C and N bisects the angle ACB in the spherical triangle ABC (a spherical triangle has great circle arcs as sides). a1, a2, ... , an is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is a. Another member is picked at random, independently of the first. Its value is b. Then a third, value c. Show that the probability that a + b + c is divisible by 3 is at least 1/4. P lies between the rays OA and OB. Find Q on OA and R on OB collinear with P so that 1/PQ + 1/PR is as large as possible. X has n members. Given n+1 subsets of X, each with 3 members, show that we can always find two which have just one member in common. 点击此处查看相关视频讲解 在方框内输入单词或词组