| 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. |
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| 2. Show that the five roots of the quintic a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0 = 0 are not all real if 2a42 < 5a5a3. |
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| 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. |
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| 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.] |
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| 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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To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
25 Aug 2002