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.
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