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The 26th United States of America Mathematics Olympiad
1997年第26届美国数学奥林匹克
  • A1. Let pn be the nth prime. Let 0 < a < 1 be a real. Define the sequence xn by x0 = a, xn = the fractional part of pn/xn-1 if xn ≠ 0, or 0 if xn-1 = 0. Find all a for which the sequence is eventually zero.
  • A2. ABC is a triangle. Take points D, E, F on the perpendicular bisectors of BC, CA, AB respectively. Show that the lines through A, B, C perpendicular to EF, FD, DE respectively are concurrent.
  • A3. Show that there is a unique polynomial whose coefficients are all single decimal digits which takes the value n at -2 and at -5.
  • B1. A sequence of polygons is derived as follows. The first polygon is a regular hexagon of area 1. Thereafter each polygon is derived from its predecessor by joining two adjacent edge midpoints and cutting off the corner. Show that all the polygons have area greater than 1/3.
  • B2. Show that xyz/(x3 + y3 + xyz) + xyz/(y3 + z3 + xyz) + xyz/(z3 + x3 + xyz) ≤ 1 for all positive real x, y, z.
  • B3. The sequence of non-negative integers c1, c2, ... , c1997 satisfies c1 ≥ 0 and cm + cn ≤ cm+n ≤ cm + cn + 1 for all m, n > 0 with m + n < 1998. Show that there is a real k such that cn = [nk] for 1 ≤ n ≤ 1997.
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