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The 18th United States of America Mathematics Olympiad
1989年第18届美国数学奥林匹克
  1. Let an = 1 + 1/2 + 1/3 + ... + 1/n, bn = a1 + a2 + ... + an, cn = b1/2 + b2/2 + ... + bn/(n+1). Find b1988 and c1988 in terms of a1989 .
  2. In a tournament between 20 players, there are 14 games (each between two players). Each player is in at least one game. Show that we can find 6 games involving 12 different players.
  3. A monic polynomial with real coefficients has modulus less than 1 at the complex number i. Show that there is a root z = u + iv (with u and v real) such that (u2 + v2 + 1)2 < 4v2 + 1.
  4. An acute-angled triangle has unequal sides. Show that the line through the circumcenter and incenter intersects the longest side and the shortest side.
  5. Which is larger, the real root of x + x2 + ... + x8 = 8 - 10x9, or the real root of x + x2 + ... + x10 = 8 - 10x11 ?
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