The 12th United States of America Mathematics Olympiad
1983 年第12届美国数学奥林匹克 
 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 .
 Show that the five roots of the quintic a_{5}x^{5} + a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0} = 0 are not all real if 2a_{4}^{2} < 5a_{5}a_{3} .
 S_{1}, S_{2}, ... , S_{n} are subsets of the real line. Each S_{i} is the union of two closed intervals. Any three S_{i} have a point in common. Show that there is a point which belongs to at least half the S_{i} .
 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.]
 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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