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The 12th United States of America Mathematics Olympiad
1983 年第12届美国数学奥林匹克
  1. If six points are chosen sequentially at random on the circumference of a circle, what is the probability that the triangle formed by the first three is disjoint from that formed by the second three .
  2. Show that the five roots of the quintic a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0 = 0 are not all real if 2a42 < 5a5a3 .
  3. S1, S2, ... , Sn are subsets of the real line. Each Si is the union of two closed intervals. Any three Si have a point in common. Show that there is a point which belongs to at least half the Si .
  4. Show that one can construct (with ruler and compasses) a length equal to the altitude from A of the tetrahedron ABCD, given the lengths of all the sides. [So for each pair of vertices, one is given a pair of points in the plane the appropriate distance apart.]
  5. Prove that an open interval of length 1/n in the real line contains at most (n+1)/2 rational points p/q with 1 ≤ q ≤ n .
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