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The 26th British Mathematical Olympiad
1990年第26届英国奥林匹克数学竞赛
  1. Show that if a polynomial with integer coefficients takes the value 1990 at four different integers, then it cannot take the value 1997 at any integer .
  2. The fractional part { x } of a real number is defined as x - [x]. Find a positive real x such that { x } + { 1/x } = 1 (*) . Is there a rational x satisfying (*) ?
  3. Show that √(x2 + y2 - xy) + √(y2 + z2 - yz) ≥ √(z2 + x2 + zx) for any positive real numbers x , y , z .
  4. A rectangle is inscribed in a triangle if its vertices all lie on the boundary of the triangle. Given a triangle T, let d be the shortest diagonal for any rectangle inscribed in T. Find the maximum value of d2/area T for all triangles T .
  5. ABC is a triangle with incenter I. X is the center of the excircle opposite A. Show that AI·AX = AB·AC and AI·BX·CX = AX·BI·CI .
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