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The 5th United States of America Mathematics Olympiad
1976年第五届美国数学奥林匹克
  1. The squares of a 4 x 7 chess board are colored red or blue. Show that however the coloring is done, we can find a rectangle with four distinct corner squares all the same color. Find a counter-example to show that this is not true for a 4 x 6 board.
  2. AB is a fixed chord of a circle, not a diameter. CD is a variable diameter. Find the locus of the intersection of AC and BD.
  3. Find all integral solutions to a2 + b2 + c2 = a2b2.
  4. A tetrahedron ABCD has edges of total length 1. The angles at A (BAC etc) are all 90o. Find the maximum volume of the tetrahedron.
  5. The polynomials a(x), b(x), c(x), d(x) satisfy a(x5) + x b(x5) + x2c(x5) = (1 + x + x2 + x3 + x4) d(x). Show that a(x) has the factor (x -1).
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