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The 2nd United States of America Mathematics Olympiad
1973年第二届美国数学奥林匹克
  1. Show that if two points lie inside a regular tetrahedron the angle they subtend at a vertex is less than π/3.
  2. The sequence an is defined by a1 = a2 = 1, an+2 = an+1 + 2an. The sequence bn is defined by b1 = 1, b2 = 7, bn+2 = 2bn+1 + 3bn. Show that the only integer in both sequences is 1.
  3. Three vertices of a regular 2n+1 sided polygon are chosen at random. Find the probability that the center of the polygon lies inside the resulting triangle.
  4. Find all complex numbers x, y, z which satisfy x + y + z = x2 + y2 + z2 = x3 + y3 + z3 = 3.
  5. Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression (whether consecutive or not).
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